Finding an apartment that accepts dogs can be a challenge here, no doubt about it, but it's generally easier to accomplish with small dogs than with giant ones. At one point, the Dane County Humane Society
Dane County Humane Society: Helping People Help Animals used to keep a list of landlords who accepted dogs, but I am uncertain whether or not they still do; one other regular Madison poster recently commented that she thought they had discontinued it.
One thing that does occur to me is
Tenant Resource Center. They operate as both a mediator between landlords and tenants, and as a sort of BBB for landlords. It is possible that they might be able to suggest some options. I've been told that their website is not terribly intuitive to navigate, so it might be best to call them.
Also, this website:
Start Renting Easy-To-Use Apartment Listings includes information on whether or not a particular apartment accepts dogs.
You shouldn't need to live an hour away from the city to have your pets, but it may take a bit of hunting to find the right spot. If you plan a visit to do some on-ground research, it might be easier in person than over the internet. But it certainly should be possible.
Good luck in your search and your upcoming move!