Handgun hunting can be done on any public WMA or National Forest during regular open hunting seasons. Check the regulations
If you are really interested in taking up deer hunting with a handgun, you have a number of important decisions to make. First on that list is what type of handgun you like the most and can shoot best. Some may consider the caliber choice to come first. There is a fine line here, so picking either first is okay.
The assumption being if you like the gun, its action, and its features, you will practice with it more. That is the key to successful handgun hunting. Unless you are willing to shoot your primary handgun for hunting enough to be highly proficient with it, then this may not prove to be your cup of tea.
The three basic types of handgun actions to consider for hunting are the single action (western type revolver), double action (Dirty Harry type revolver), and the single-shot bolt or break-open handgun. I am biased, but for handgun hunting, forget the semi-auto type action. The exception might be a Desert Eagle in .44 Magnum.
Each type in any given firearm of high quality and known reputation will be fine for deer hunting if the hunter will learn to shoot it accurately at a variety of deer hunting ranges, usually 100 yards or less. Handgun hunting is generally a short-range affair.