This is a loaded question but I should set out to define competitive and casual.

Competitive would look like Modern where there are many events, both paper and digital. There are defined decks but new decks are always entering the format.

Casual format looks like Old School. The format is mostly solved, the diversity of decks is usually based on creativity not competitive play. There are events but they are not as frequent or predictable.

I am not sure I will get much response to this thread and that is fine. But, I was speaking with my brother about preparing for upcoming events and we were discussing this very topic. We both play Vintage and Old School but based on the amount of events available right now. Vintage is feeling more casual and not as competitive.

I like to think of Vintage as competitive and maybe it is. Maybe it's mostly competitive online? I don't know, I am asking the question.

Its a combination. There isn't a Vintage pro tour event so obviously the pro players impact is much lower than in formats where there is a pro tour. As such, I think most people are playing the format for fun. That doesn't mean that they aren't going to be competitive and try to win though.

There are some players that will switch decks often, and constantly trying new things to get a leg up on competition. There are other players that will run the near same 75 for years, and/or run decks that they know are sub-optimal just because they find them fun.

There are areas of the world, where tournaments are frequent and predictably held (like online and in the North East), and there are areas, where they are not.

The format doesn't rotate, or have frequent bans (like Modern + Standard) so the format is "solved" to some degree in the same way that Legacy is "solved". We obviously still have the occasional restriction, or major printing that will change the format.

I think Vintage is a competetive format in general.

Its not solved, its constantly evolving, the play patterns are very skill testing in ways that are not common in other formats. What i very much like about Vintage is that often every small little mistake or misjudgement can lose you the game on the spot. In other formats, while this can certainly also happen, especially against combo decks, it is not nearly as common, and most small mistakes just lose you value or tempo. (which ofcourse might lose you the game down the line, but rarely immediatly). For the record: Iam not trying to say that other formats are less skill intensive, it is just different kind of skills that are commonly used in Vintage.

It is also not an easy format to get into, especially for new players. Most new players can easily pick up limited, standard or even modern. Not so much with the older formats, and while price and exposure play a role in that for sure, i think it is just hard for new players to grasp the huge amount of mechanics, legal sets and cards as well as play patterns and rules (a lot of strange triggers, interaction on the stack, stacking of abilites...). The format also really rewards experience. For those reasons i believe Vintage is very well suited as a competetive format.

Why then does it feel so relaxed? I think its a number of reasons:

Some players still think that Vintage is more luck based than other formats ("whoever begins wins on turn 1", "hey had the force, they just got lucky", ...) and i think this stigma is just plain wrong. Ive been playing standard for a while now just because i can play it for free on mtgA and there are many many games that feel to me, that it was just luck based and as a control deck player i often feel 'bullshitted' when my opponent begins and plays threads ahead of the curve before i have enough mana for countermagic. This feeling might just be because i dont understand the format enough (however iam not really interested in it enough to do so). If i play older formats like Legacy and Vintage i very rarely feel like it, one one hand because you have cheap and even manaless interaction and on the other hand because i think i understand and know the format way better and can predict when a turn might get very explosive, what too look for and how to play around it. (at least thats what iam telling myself, in reality i dont even own power...scub). I think understanding a format and variance greatly reduces the feeling of "i just got unlucky, its just a coin toss".

The next reason might be the lack of pro or semi pro interest to up the average competition because of the lack of regular high level tournaments and WotC support. This got a bit better with the introduction of the format points of mtgo and look how a great player like Nassif ran through the field (iam aware this might just be a statistical anomaly, but it is impressive for sure) . Pro(-team) interest would probably also speed up the evolution of the format, if that is a good thing everybody has to decide for themselves.

The next reason is i think is very positive in my opinion, but still contributes to an over all more casual feel: The player base of Vintage is much older compared to other formats. What does that mean? Players that, while still definetly wanting to win and beeing very good, tend to be less spiky. Most players dont throw fits or get overly salty when they lose (something that is very commonplace in standard and modern). I think the Vintage playerbase is just more level headed, mature, releaxed and way less toxic than the average Standard or Modern player. Players are more trusting and know how to have a fun night, while still playing competetive and highly skill intensive Magic. That attitude might seem less caring, but in my experience people care about this format a whole lot, they just dont scream and kick to make their voices heard like it is sometimes commonplace in other formats (Well, at least not outside a B&R or Reserved List thread )
Toxicity does not equal competetivenes per se!

In summary Vintage as a format is potentially competetive and very skill intensive, but the players tend to radiate a more relaxed attitude, and some misconceptions make the format seem like something it really isnt.
Thats just my thoughts about it.

Its competitively casual. One thing we all need to accept is, Vintage is not truly supported by a Wizards. Nothing props it up. Competitive events rarely occur. Cards are not printed for it. Those who oversee Magic really do nothing for it. The NE meta (eastern Pa, central/northern NJ/NYC) is considered one of the most active and biggest meta's in the world for paper. Honestly, there are may be 70 unique players.
Deal Me In Games last year had 11 events and I think 60-62 unique attendees. Top deck, beareded dragon, and the comic store in NY (jp or alan often posts their events, I apologize I can't think of the name right now) pretty much share those 60 unique players.
The NE is the last bastion of multi-state, multi-store tournament scenes and its not even sanctioned. If we get 20 players we are happy. If we get 25 players we are ecstatic. If we hit 30 players we hit numbers not seen for 1 1/2 years.
There are small pockets of vintage communities (shaman ben's in the mid-west, the Austin, TX events and I thinks Southern California). But let's face it, those 4 hold outs are small, unsanctioned, usually non-judged (in that no judge is hired and paid for officially, though occasionally it happens) with the same few people at each. The events are store run, usually without support from Wizards. Everything about that screams casual format.
With that said, my initial claim is that we are competitively casual. The players who come, come prepared. We have fun, we are often easy-going with each other, but we fight and fight hard when we sit down to compete against each other. We take our deck building seriously. We take our cards seriously. We want to win. But we are looking to win in a casual format.

Note for reference - I have 0 online experience so that may be a different story.

This was the subject of an article I wrote in 2003:

Old Format, New Conflict: Old-School Versus New-School In Type One

Vintage always leaned in between those lines. You cannot judge a format's competitiveness out of its interactions IMO. I played a bunch of 1x1 Commander over the years and that format is NOT competitive - but has all the qualities people here think competitive formats have. I for one believe that if you never take out your DCI and if we almost never have official supported tournaments, you cannot call a format competitive. It's mostly casual.

Have in mind that all of this could have changed this year the moment Wizards announced Vintage can be used to qualify to the Pro Tour. I guess that bumped a bunch of casual formats to competitive status.

@smmenen It's a good article and it touches on part of my question but does not arrive at where we are now, obviously because we are 16yrs in the future.

I asked this question because a casual format can be competitive. And a competitive format can be played in a casual way. I pushed myself to be much more competitive than I have been in some time and I enjoyed some of it and hated some it.

I do not long for decks of yesterday and I do not miss the old meta-games either. I miss being able to play paper Vintage every Monday. I miss having a somewhat large group of players playing the game regularly.

MTGO offers some of this but it is not the same. I am still trying to figure out how to use MTGO in a way that I find satisfying. I am still trying to figure out if paper Vintage can survive and grow. I have ideas on how to grow it locally but I am middle aged with 3 kids, a wife, and a mortgage.

Vintage is competitive and it is casual. That's just where we are now.

@moorebrother1 I think as long as there are championships, it will be competitive. There may be players who like to go to championships to play casually, and there's nothing wrong with that. But there will always be people playing it competitively. Even commander 1x1 can be competitive. I knew some people that had some pretty expensive and well built x1 decks around here, and there was also the league on MTGO.

I think they are 2 things: the incentives for people to take the format competitively (PT, GPs, even MTGO Leagues, Format Championships, etc) and the people itself. The format doesn't give by itself incentives to become competitive (well... not for me in Brazil anyways). But there are competitive people playing it (as there are in almost every format, I guess).

@gutocmtt the fact people treat the format competitively doesn't make it competitively IMO. But that depends on defining the term, I guess.
For me it's like: what incentive a profissional player has to play that format and take it seriously and train for it? None? Then it's not competitive.

I also played a lot of Commander in Brazil (we probable know each other right?) and although a lot of people treated the format in a highly competitive manter, I'd never call it competitive, not even in its glory days. The most you could do with your Commander "grind/career" was win some local tournaments and maybe a "national" that only contained people from 1/20 of the country once in a while.

The most concise way of answering the question would be that it is both. This is part of vintage's appeal IMHO. It is at the sweet spot between both sides of the spectrum.
It is a skill-intensive format and players are competitive but at the same time, the lack of tournament support and number of players gives it a casual ambiance.
It also depends on the platform. As far as paper is concerned, it is much more community oriented. Whereas MTGO would be the competitive expression of the format. This is clearly evident in metagame differences where people tend to play what they enjoy in paper tournaments and MTGO tends to feature the tier 1 decks in larger proportion.