Firstly prom dates aren't always because you like each other as 'more than friends', therefore so what is she says no, you can say you just wanted someone to go with! Doesn't mean you're pouring your heart out to her does it, because like you say you aren't that fussed about her! (or are you secretly?)
I'd just do whatever feels natural, personally face to face may seem the most awkward, but I think it'd work out easier if you did it that way, just drop the prom into conversation and mention you don't have a date, if she says 'no, nor do i yet actually' or something similar then drop the question in.. only if she changes subject would i even hesitate to ask.
Although seriously, "if you don't ask - you don't get."
You only live twice mate
