Will I get in trouble at camp? Or teased?

No you probably won't get in trouble but possibly teased or gossiped about. Don't feel bad or embarrassed you mom just doesn't want to lose her little girl. Ask your friends if you could borrow a bra if they have a spare if you can't convince your mom. If non of your female friends are going try and make friends with other girls maybe one of them will let you borrow one for the trip.