Unless the bottom really falls out of things in the next six months or so, I don't think you'll have any problem at all in finding an apartment in Madison. There has always been a plentiful supply of rental housing here, and a fair number of landlords have built new complexes in the past 6 to 8 years that have increased the supply even further. The demand has stayed fairly steady, and as a result, many complexes have a consistent vacancy rate of 5% to 10%, and in some areas, the vacancy rate can be much higher. You can even find some really sweet move-in deals that landlords offer to try to attract new tenants.
Probably the single thing that could make it most difficult to find an apartment is the fact that you're moving in December. Most folks hereabouts try to avoid moving during the snowy months, for obvious reasons, so the availability rate in December and January is likely to be lower than it might be in, say, May and June. That said, you are still unlikely to encounter too much difficulty in locating a place, unless your requirements in terms of size or amenities are extremely unusual or esoteric.
Do make a point of checking in with the friendly folks at
Tenant Resource Center when you're closer to the point of atually hunting for a place to stay. They can tell you the good, the bad, and the very, very ugly of Madison's rental market, and anyone who'll be renting in this city should check in with them before signing a lease. There are some very fine landlords in this community, who genuinely treat their tenants as respected clients and their properties as assets, and then there are the others. The Tenant Resource Center can help you find out which is which before you sign a lease.
Good luck! Check back in when you're closer to the point of moving, and we'll be glad to offer any help we can.