I need to have my dog shipped by air from oklahoma to washington state. what companies do this? -

Try the airlines with direct flights between the cities.

I used to work for United Airlines and they did it. But I have to tell you that any animal that has to ride in the baggage compartment of an airplane gets very stressed out. I would see if Amtrak will do it.

Most airlines will do it and there are also pet transport services that do it. See list below.

UPS.