What section are you doing?
The responses above are for the wilderness section - typically ~2 nights and ~38 miles from Cooper's to Big Eddie. It's the premier section so that's why folks assumed that's what you were asking about. Up the Creek Shuttle is the same as the one Cooper will hook you up with, I've never needed any reservation as they do a lot of shuttle business.
There is also a "day run" section from Chavez Canyon to Big Eddie where a dirt road follows the river, I've seen people bike shuttle that, but I prefer to just hitch a ride if I can't self shuttle that section. That section has the majority of the rapids and requires no fees or permits with ample free camping, and is the last 8 miles of the typical wilderness run (although you can take out from the wilderness run at Chavez Canyon if you prefer, the shuttle then costs ~$120). There's no shuttle for hire in the day use section that I'm aware of.
There is also the "Mama Chama" run from Rutheron (NM95 Bridge) to El Vado, which is what I'll be running tomorrow. I don't know of a shuttle for that section, it's about 15 miles with a paved road that roughly follows the river, you could bike shuttle that, but 15 miles in a kayak then 15 miles on a bike sounds like a very long day to me. I don't know of a shuttle for hire on that section.
Have fun - there's plenty of water in all that right now!