I don't know of anywhere that supplies veneer sheets in the Birmingham area as such. However, you may find that Midland Veneers, who design and apply their own veneer products, might be able to let you have a sheet. I've never dealt with the company myself, but there'd be no harm in asking.
Silvermans have quite a few branches around the country. The flkexible veneer that they do is already edge jointed so you just cut what you want and stick it on.