Is there a simple system for an end user to install on a website so a business can enter in a list of their members and then be able to give out access so other members can search and connect with other people from the same zip code?
I am not sure what exactly you are looking for. But chanced are rare for such a script.
But you can easily code this using php. Just have to insert the users with their zip and other details and while searching just do a mysql query using the LIKE command or a WHERE "=" clause.
Such systems do exist - a good place to start looking is probably http://php.resourceindex.com/ (or pick another scripting language of your choice) - you can't easily code them in PHP if you don't know PHP!
The issue with finding users that are "near" each other is that you have to map zip codes to latitudes and longitudes - this is called geocoding. Once you have latitudes and longitudes you can use something a bit like Pythagoras theorem to find those that are close to each other. Google maps has an API that will allow you to do the geocoding and there's probably scripts on resourceindex.com that will do this for you.