With Alamo, when I book through Costco, they give me a Costco booking # and an Alamo booking number. I just go to the Alamo site after making the reservation and from their actual website I complete the form to skip the counter and print it out.
So budget is probably similar in that once made through Costco you just call and they'll add the other info onto the reservation for you.
Edit: Slickd23 posted while I was and knew the facts about the Fastbreak number, lol. The free 2nd driver Costco throws in is well worth it when their price is similar to elsewhere using coupon codes. Alamo wants $10 more a day for an extra driver without using Costco, ouch.