Is it safe to assume that a bicycle is also classified as a vehicle since they have to pay attention to the rules of the road? I would also guess since a lot of us ride trails where bars are stops along the way where one could potentially get ticketed for OWI or DUI if they are classified as a vehicle.
Since trails go through different counties, local laws would change each time you entered. What a crazy world we live in!
Yes, you absolutely can get an OWI on a bicycle. But it isn't heavily enforced on trails unless you create an incident.
As far as local laws, I'm not sure there are many that would supercede state law.