Is it necessary for a vegan who wants to build muscle to take a protein supplement, or can they get enough protein from diet alone?
To be fair, it would be very difficult. The challenge is to keep protein up, but not overshoot your overall calorie requirements. Many vegetarian bodybuilders are pesca-vegetarian (eat fish) or ova-vegetarian (eat eggs), neither of which qualify for a vegan diet.
You want your macros to be somewhere around these numbers:
Muscle won't grow faster than 3/4 a pound a week, and that is a reasonable target to keep from adding too much fat.
Now, there are several articles out there that can help you make some good food choices. A good starting point would be "Vegetarian Bodybuilding". Unfortunately even in this article it does suggest considering going lacto (milk)/ovo-vegetarian as both of those have good sources of protein (particularly yogurts and keifer). There's some good options for you in that article that would help you hit your goals with food.
The problem you'll run into is that plant based meals are typically high in carbs and fiber, which can actually cause you to not get all the Calories you need to grow muscle. Probably the best strategy is to break up the meals into several smaller meals until you get all of the required food in. You'll also want to incorporate foods with higher fat to help the the Calories up, such as avocados. If you consume plenty of soybeans, you might be able to hit your target macros decently.
Is it possible to build muscle without supplementation on a vegan diet? I suppose anything is theoretically possible. But is it going to be easy, or even practical? No, despite what some vegans might claim. The quantities of protein needed for muscle building, especially at the bodybuilding level, is just too high. Even non-vegans have a difficult time achieving their protein goals, and that's with animal proteins and supplementation.
Now, as for supplementation, there certainly are ways to do this while adhering to a vegan diet, such as with soy or hemp based protein powders, rather than whey. So I don't see why there should be any objection on the part of vegans with the above. It's not like I'm saying you can't be a vegan and build muscle. I'm just saying that it's not realistic to do it without supplementation--as I said, even non-vegans have a really hard time with this.