I'd like to promote my offensive co to Head Coach. Is this possible? I can't see where to offer him the contract. Do I have to fire my HC first? If so how long would it take to get in a new OC?

Yes, you can just offer your OC a new contract as the HC, which will in turn fire your HC and open up a vacancy in your OC position. The contract offer button is visible on his coach card when you click on his name. The one exception (and this is probably why you don't see the contract button) is if he is in the first year of his contract, you will need to wait until the offseason to make him a new offer.

Does that mean I have to go a whole season with my current head coach?

No, you can hire a coach to replace him, you just can't promote one of your coaches that's in the first year of his existing contract.