I am a vegetarian and interested in gaining muscle, bone weight, and other non-fat mass. I do eat eggs but I plan to stop eating that too and turn to complete veganism. If I exercise less and eat more definitely could I gain some fat and muscle?
Gaining healthy weight is the same for vegetarians as it is for omnivores: lift heavy, eat big, prioritize getting bigger. The only difference is that vegetarians and vegans make it harder for themselves by not eating (or eating less of) some of the most useful foods: meat, dairy, eggs. These foods are great because high-quality versions of them contain good fats, lots of easily-absorbed protein, and they are nutritionally dense. This makes the difficult process of eating enough a little easier. If you refuse to eat those useful foods, then gaining weight is possible but necessarily some amount more difficult.
There are no special solutions. You'll still have to find a way to lift heavy, eat big, and prioritize getting bigger. You'll have to find a variety of healthy foods that get you the protein you need, while not making you so full that you can't eat enough total food. My advice is to at least eat ethically raised farm eggs and dairy alongside a variety of vegan protein sources like beans, soy, and so on.
As protein's a major requirement for bodybuilding vegetarians and vegans will naturally place a limit on the variety of nutrients available to them. I get my protein from multiple sources, meat, fish, dairy, supplements etc. That said it's not impossible to gain mass, you just need to seek out the right foods, but I would expect it would likely take longer to achieve results.
Soy has a mixed reputation as a good quality protein source. Some folks in the bodybuilding community don't rate it, I do, but I have it as part of a mixed protein diet.
Quorn is good, I prefer the meat substitute over the poultry.
Other things to consider - Nuts, nut butters, seeds (hemp & chia), grains.
Dairy milk substitutes: almond, coconut, rice & soy.
Supplement wise, there's quite a few to go at but I like pea protein the best, by quite some distance. Other notables are rice, soy and hemp protein powders.
Don't forget it's not all about protein for gaining mass, good carbs & fats are just as important.