How can I keep my dog from getting covered in mud in the yard?

Can you fence off a small part of the yard so the dog can still go out but the whole yard won’t get wrecked?

Fence off a section and cover that section with gravel. Use only for a dog run and clean it every day. Make sure there is a shaded spot, a dog house and fresh water out there too. Or you could use pavers instead of gravel. You would, of course, get permission from your landlord first. If they don't agree explain the problem and ask them for solutions they find acceptable.

Was this issue not considered before you rented the house and signed the rental agreement? At this point I would contact the landlord and see what you both can agree on. My thought would be to install a temporary dog run and use a tumbled pea gravel or pavers. Then agree to restore the area to its original state - at your cost - when you move out. The landlord will most likely be grateful that you don't want the dog tearing up the beautiful grass over the winter months. Or make a dog cleaning station by the door that the dog uses. I used to have to do that with a friends shepherd right after they purchased the house for the first winter. While it wasn't exactly convenient, it was doable for one winter.

is the front yard any better? you may have to take your dog out on a leash

You can't really put anything over the grass or the grass will die. You could make a temporary enclosure to limit the dog to one area. We use metal fence posts and snow fence held tight with wire ties. Limiting the dog to one area really isn't going to keep the dog clean. One of the cons of owning a white dog! Lots of baths!

Walk your dog, great exercise for all. That way you can control where your pet is.