Well have you ever shopped for underwear; it's kind of the same thing. Like autumn said everyone goes through this. If you want to fit yourself and buy them yourselves the cashier could care less. Even though the question seems embarrassing to you it shouldn't be awkward for your mom at all. No more awkward than asking for sanitary pads right?
Just tell her your getting older, and as you get older you need things. In this case you need a bra. Don't feel nervous about it, every one goes through this.