Can you bring you're own switch to play a game example Mario cart?

Yes. Both places allow you to bring a Switch and pay to use a TV.

Yes you can bring it into both from my understanding. They have 2 TV's in legendary and a ton in alpha.

I think you're asking about Alpha Esports Lounge. This is the venue that does video gaming. Yes, you can bring your own Nintendo Switch to play games. Legendary Wolf Games is a gaming store for board games and trading card games that also hosts tournaments to play games.