I'm vegetarian bordering on vegan, for environmental reasons (and the simple fact that I just don't like the flavor/texture of meat, particularly beef). My spouse and kids consider themselves flexitarians -- primarily vegetarian with an occasional bit of meat here and there. Both can be as expensive or as cheap as you like. When the family went flex several years ago, our grocery budget went down by about 10%. With two teen boys, that was no small amount.
If you eat a ton of processed food or fast food, vegan or otherwise, it is going to be expensive in both money and health. The only specialty items we get are nut milks, but we have two lactose intolerant people in our fam so that would be the case regardless. Sometimes we make our own. If you eat out of season, it's going to be expensive. I prefer scratch-cooked meals at home that follow the basic recipe of veggies--carb--legume. Stirfries, pasta, flatbreads and wraps, soups, chilies, etc. These are cheap no matter how you like them, even cheaper if you leave out things like meat and cheese. Our meat "substitutes" are cheap, too -- a $1 block of tofu feeds a family of four in a dish. A pound of TVP (I use this in pastas or to make a sausage substitute) is $2/lb from the Winco bulk bins (and it doubles in size when reconstituted), and vital wheat gluten (for making seitan) is also around $2/lb in the bulk bins. Dried beans and legumes, of course, are also cheap.
As for eating out, we tend to opt for Asian takeout, particularly non-chain places with cuisine from Buddhist-heavy countries. There is nearly always a vegetarian or vegan option that cost the same or less than other options. In our area, at least, burritos and tacos are also usually a safe bet at the local chains, with most places having true vegetarian options that don't cost more. And of course, a vegetarian salad is a possibility just about anywhere (and salads are often less expensive "add-on" items, anyway). The Happy Cow app is great for finding places with a decent menu in your budget. Now, if you want to eat out at a McD's or other burger joint, your options will be limited and any veg options will be priced at a premium and weak at best. But really, why would one go to a burger joint if they are vegetarian? It's like going to a sushi place for the ice cream. When traveling, if there is a McD's then there is likely a Subway or similar sandwich shop nearby, just go there instead.