The Car Care Center and the Dolphin both have Alamo and National rental counters. The CCC location will send out a van to pick you up at your hotel, and then drop you off when you return it. It'll even drop you off at one of the parks, if you want. (I've heard that they can arrange to have the van service for the Dolphin location if you really need it, but I have no confirmation of anyone using it)
Typically I've found that Alamo has the better rates, but it can be worth it to shop both.
If you want to use another company, some of the hotels along Hotel Plaza Blvd (near Downtown Disney) have rental desks, but I don't know if there are shuttles available. If it's a nice day, you can use a Disney bus to go to the Marketplace and then walk from there. (The farthest hotel is the Doubletree, and that's about 8/10 mile from the Marketplace bus stops)
Alamo/National has an additional location at the Buena Vista Palace.
Avis is at the Hilton.
Budget is at the Doubletree.
Dollar is at the Regal Sun (formerly the Gosvenor).
Enterprise has a location near Disney property. It's not walkable, but they do advertise their "we'll pick you up" on TV all the time.
If you want to rent the car in one location and drop it off at another (like the airport, or if you want to rent at the CCC and drop off at the Dolphin), it's Florida law that they cannot charge you a drop fee or for a one-way rental, as long as the pickup and drop-off are both within Florida.
Check mousesavers.com for discount codes and coupons. If you belong to AAA or a warehouse club like Sam's Club or BJ's, they often have discount codes and coupons, too.
Lastly, after you make your reservation, keep checking back on the website every few weeks. You'll often see prices drop, so you can then book the new rate and cancel the old one. (I've been able to drop my rental with Alamo by 30% because I kept checking back, and it'd go down $8 here, $15 there...)
-Rob