You can *do* it (particularly if you prime and paint it real good -- be aware OSB uses a LOT more paint than plywood, because it is so rough, and you really need to get it *all* covered, and in several THIN coats not fewer thick ones, too. Also make REAL SURE that the bottom parts are not gonig to be EVEN REMOTELY in ground contact, nor embedded in wet weeds half the year, nor having wet chicken-run footing piled up against them; because even if it's painted, OSB does not have a great sense of humor about prolonged dampness.
Whether it's a good idea depends on how long you want your coop to last. With minimal roof overhang and no paint, it may start becoming dangerously weak (from a predatorproofing viewpoint if nothing else) within a few years; with large roof overhang and a GOOD coat of paint you are probably good for 10 years; but plywood would last better and longer.
Good luck, have fun,
Pat