First - since you can't fold your wheelchair, a car will probably not help much. My daughter has a Quickie wheelchair with a rigid frame. In order to get it into a car trunk, we usually have to pop off at least one of the wheels, take off the seat cushion and back and then somehow make all those pieces fit in the trunk. If your friend also has an
ECV, that would be another thing to take apart and get into the car.
Wheelchairs are generally not folded on the buses - they are fastened down with tiedown straps on the floor of the bus. People who fold the chair are usually those with very plain rental wheelchairs. Folded chairs need to be help on the bus, so that would not work for you anyway.
Wheelchairs and
ECVs load and unload from the back of the bus, at a ramp that the driver puts down so you can roll in and out. The dog will be able to walk up the ramp easily.
Here's what I would suggest (my daughter has a service dog, but has not brought him to WDW, so those with experience might suggest something else).
Have your friend with the ECV load onto the bus first. Once the driver has gotten her ECV attached down, send the dog in to her and have her hold onto the dog while you get in. It is recommended that guests with 3 wheeled ECVs transfer to a bus seat because the ECV will be much more stable on the bus without anyone on it.
You may want to practice sending the dog to her before you go and also after she has gotten the ECV. Those things should not be a problem for a service dog, but they are strange and added the sights and sounds of using a bus and ramp to what is going on, it would probably make your dog more comfortable to have some pieces of the process familiar.
It will take a few minutes for the driver to attach the tiedown straps to your wheelchair and fasten it down. If you don't have a transport option with installed tiedown loops on your chair, it will be faster if you can figure out safe tiedown points (strong frame pieces, not anything bolted or attached on). Colored duct tape or ribbon works well to mark. Then you can tell the driver he/she can attach anywhere they see the tape or ribbon.
Your friend should hold the dog while your chair is being fastened down, then you can call the dog back to you to settle close to you.
When getting off the bus, you would get off first and then call the dog down to you before your friend gets off.
A couple of other things to be aware of -
WDW still has some older buses with lifts, although these are being phased out. You do not want to use those, so if that kind of bus shows up, just let the driver know that you need a bus with a ramp and to please call for one.
There are attractions where Service Dogs are not allowed. For those, you will have to provide someone to hold the dog for you while you ride. If it's just you and your friend, you will need to trade off riding and holding the dog. Those attractions are listed in the regular park maps, plus there are special park maps for guests with disabilities that have more information.