1. Finding a veg-friendly club is pretty much going to be impossible (best most can hope for is "veg-tolerant"), so all you can do is show by example and make people second-guess their anti-veg views by giving clients the results they're after and putting the naysayers to bed. If you work for someone else, you're going to have to play by THEIR rules, so it's one of those situations where, if you can't handle the fact that it's likely that others you work with will try to put your lifestyle down in order to make themselves look better (such as, other trainers who skirt by without doing much work and get lackluster results for their clients), then it's going to be difficult unless you get lucky and end up somewhere small with a very understanding staff who isn't out to question veganism at every chance. Some vegan trainers have probably ended up in great situations early on, but there are no shortage of stories of people who were told that their diet was poor, that they couldn't push clients to eat meat-free, etc., so you'll want to ask a LOT of questions prior to taking a position, just to be sure you'll be the right fit for working there.
2. The MOST vegan-friendly way to train will be to do training yourself without working through someone else's operation. This will require more work, but you won't have to deal with any of the b.s. from co-workers who will try to convince others that you can't do a good job being vegan, that you're screwing up clients by pushing them toward a vegan diet, etc. But, not everyone is cut out to be their own boss, and there's a LOT more work to do when you have to handle everything from the setup of your LLC to managing finances to negotiating leases for your space to doing the training of clients. Plus, there's the $$$ factor involved in it, but there are ways to keep things cheap (such as, putting together a "travel kit" of fitness equipment so that you can do in-home training for your clients, thereby eliminating the need for a studio to train in), so it can be done, but it takes a bit more creativity to keep things cheap while being your own boss.
Can't offer much more than that - once I got my ISSA certification, I ended up working in the world of vegan products and never actually ended up training anyone, but definitely looked around at facilities and spoke with a LOT of people during the time I was considering being a trainer. Many of the guys I've lifted with over the past decade have been trainers themselves, so I've had a lot of info from those who have been in the trenches at cruddy gyms as well as those who work in college athlete coaching settings, so I've definitely heard a lot!
"A 'hardgainer' is merely someone who hasn't bothered to try enough different training methods to learn what is actually right for their own damned body." - anonymous