Can I mod my team competitively with the modules store?

I got my best mod, which is a 17 speed secondary from the mod shipments (speed secondaries are extremely important, as well as speed primary arrow), so its definitely possible. If you are running droids you ideally want to be getting critical damage mods to increase the damage your droids do.
Don't expect to be able to buy the mods from there for your entire arena team though, you will have to get those elsewhere. If you can afford them, you could try buying the mod packs when a tournament is on.