Tom, I've actually shot in quite a few zombie action pistol matches and one zombie all gun match (zombie 3-gun). That's actually the reason I bought my Mossberg 930.
The guy that started it is WAY to old to qualify for the "me first video game players" as it was so well put above. (no offense if you read this, John!)
They are all about fun and getting people into the shooting sports. The main idea is to use the zombie pop-culture theme to get people into IDPA style and 3-gun style competition without all of the equipment restrictions and draconian overseers. And yes, I have shot matches outside of ZSA, so I understand the idea.
Equipment is pretty much whatever you bring. There are classes by sight type and by magazine capacity.
One nice thing about the ZAP matches is that you can use whatever lights/lasers/etc.. you would normally use on your carry rig or whatever. Things that are not allowed in IDPA/IPSC leagues becuase they offer opportunity for unfair advantages.
When I did the ZAG match, there were quite a few Saigas as well as Mossbergs, Benellis, and you can even find a pump action at these matches.