First of all: USPS is NOT the same as UPS. They are completely different companies and have nothing to do with each other. (It is not immediately clear from what you write that you are aware of this.) Then to the UPS: There are two modules for UPS that are obvious choices - the UPS module that already comes with ZenCart, and the UPSxml module that can be downloaded via this site.
My experience is that UPS needs some days to activate your account for API access. So when you ask them to enable your account for access from ZenCart, count on having to wait some days for things to start working. The regular UPS module doesn't connect to the UPS server, it seems, so for this you need the UPSxml module.
I only have experience with the UPSxml module, and the experience is not at all good. It seems that the latest version is from 2006, and as far as I can understand, nobody supports it any more. For me it works now, but it gives prices that are very different from the prices that UPS are calculating for the same deliveries on their own site! I am at a loss about what to do about this myself, really.