Hedgehog, yes you definitely need to go ahead and split your log into staves. Seal the ends well with shellac, glue or paint. I assume the log was recently cut. If so, the bark will be tight on the log and you'll have to get it off with a draw knife or similar tool. If the bark is really thick, I have trimmed some of it off with my band saw, but be careful and make sure you don't cut into the sapwood. The first ring will typically be the back of your bow. I have made bows from hickory heartwood from large trees but the sapwood is generally considered the better bow wood.
Insects can attach the bark pretty quickly in warmer weather. If you need to store the staves outside, I would remove the bark before spring or spray the bark with an insecticide periodically. Borers will ruin a good stave in a hurry. Perhaps over the next few months you can get the bark off and move the staves indoors for further seasoning.
Smooth bark hickory, ie. mockernut, pignut bitternut etc. is usually considered slightly better than shagbark hickory for bows. You may not be able to tell much difference though. I have worked both. Good luck. Hope that log will yield some nice staves for you.