A quick Edict, Kotobank, Excite.co.jp, and Goo search on ガム shows that it is an abbreviation for チューインガム.
Google results for ガム also seems to suggest so. (Notice that the results only show チューインガム and no 風船ガム)
But can ガム be used to mean "bubble gum", or is it actually already a codified term for "chewing gums"?
Take for example we have both a chewing gum and a bubble gum on a table. Will "ガム持ってきて" be ambiguous because ガム can refer to both items? Or would it non-ambiguous because in that context, ガム will most surely be referring to the chewing gum?
Note: Bubble gums can be blown, like a balloon coming straight out of the mouth. Chewing gums can't achieve that, no matter how hard one tries.