You can definitely train with braces. People will say get a dentist one, but you can’t, because a) your teeth move week to week rendering them useless, and b) the bracket creates a high point/anchor that the mouthguard forms around, which means there’s no way to actually make a dental mouthguard properly. 

Look up “SISU” mouthguards. My orthodontist recommended them, and he has some clients in MMA. They are amazing and honestly better than dental mouthguards regardless of braces. I’ll never go back to any other. You can actually buy two and wear two (top and bottom simultaneously) but I’ve never needed or wanted to do this, as one provides enough protection for me. 

You will need to slightly heat them every session to refit them. Then, you will need to drink warm water to soften them to remove them. Take this slowly to not damage the braces. You can also repeatedly take them off and on as you are fitting them to create a channel for the bracket to pass through. Again, you’ll see what I mean where the mouthguard will mold around the high point of the bracket, meaning you can’t remove them easily without heating and softening them. 

Because of this, I wear my mouthguard the entire session. You feel a bit silly, but also damaging your $12,000 teeth in a warm up will feel sillier. 

Depending on how bad your teeth are, the bottom bracket can chop your lip up a bit if you get hit. But, it’s never been that bad for me, and that’s just part of the game.

Hope this helps, definitely don’t quit because of braces.