what category? in Any%, 100% no. post game category xyzabsaudh? probably.
personally, ask in the game's forum. although I'd rather not get involved with the poke-guys.
@SpiderSponge Pokemon Bank is a storage service for multiple pokemon games, there can't be a speedrun of Pokemon Bank the question is if the storage service would be allowed to be used in a run, as you can bring in High Level pokemon into the run, cutting down on exp-gathering. As I've said in my first reply - for any% runs or any other run that starts from a new file, no, it'd not be allowed, as it would bring outside material into the run. For runs that start from postgame, it might be allowed to use, since you're already supposed to be at postgame, after all.
I don't think any pokémon game offers categories that start in the post game. I don't know how the challenge mode in gen5 works, but all other categories start on a fresh file. There is no rule like that yet, but I'm pretty sure using any applications outside of the system settings and the game itself on the console is not allowed and multiplayer features have to be either turned off or ignored completly.