Bikes can come along with a character in this world, yes, given they were close to their bikes upon disappearing from their world. Otherwise, they may request it to Jessie. Motorbike could as well, but as there is no appropriate fuel an alternative would have to be developed.

Bikes can be found and bought in the Imperial City of Hera. Because the city is off-limit, they can find some smuggled bikes in Aerilon, for impressively high prices.