AFMBE is a good recommendation. You could pretty easily home-brew a system to add this on to Savage World or other games as well.
A few tables for various common areas to represent scavenging results over time, with associated tables for risks. So an RPG party might say they're going to explore an old supermarket and the GM would roll on a table for a supermarket that is heavy on canned food, etc. with a rare chance of finding a corpse that might have a weapon. However, the supermarket risk table might have a pretty good chance of zombies, possibly some weird stuff like fungus monsters if it's a setting where that makes sense.
Scavenger an 'industrial building' would likely result in tools and such with a much lower risk of zombies, but the 'risks' chart could have some related industrial risks like electrocution, misfiring machines, etc.
Scavenge a police station and you might find some goodies but the risk is also high...
Basically, the tables give the GM some 'elements' to play with. For a system that is minis-based, the GM can then sketch out the area, place the goodies and the risks.
Actual play would be the players deciding where to head next, and the GM throwing appropriate challenges at them. The players can scavenge, or instead focus on fortifying a safe spot or other related tasks.
Most GMs would get bored of this kind of play, but locations could be loaded with plot hooks to lead into some sort of grander storyline beyond basic play. Maybe each table has a 'GMs choice' option so the reward for scavenging a site might be a map to a higher risk/higher reward cache someone placed, a lead to find other survivors, transport to safe zones, information on the zombie plague basis, even a cure.
AFMBE, from what I heard, has some 'plot hook' stuff built in to character generation. Methodical foraging and looting tends to get old quickly... But knowing that Player A wants to head east to find his little sister, while Player B thinks there might be an immunization to the Zombie Plague somewhere on the old university campus North, and Player C just wants to find a nest of them and go out in a blaze of glories because his family's already been eaten... That's when it gets interesting.
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