We are relocating, moving across country in a couple of months. We have a horse to move and are unsure as to whether we will have to fly him there. We have just short of 2000 miles to move.

Could a horse cope with that in a trailer or should we arrange to fly?

A horse could cope with that in a trailer, but when you stopped overnight at a motel he/she would probably appreciate a nap in a real stall as well. There are plenty of show families who follow the circuit from Florida to California each year with multiple horses.

Are there many motels that have stabling facilities? Will we be able to find something like this relatively easy, or easy enough that we can make it the full distance?

I have never seen one that does but that doesn't mean they don't exist. Its possible you could look online for bed and breakfast type places since I know there are some in the country (not smack dab in the middle of town/city) that will allow so.