No, they don't. The NSA doesn't do this directly, they use their legal authority to require that providers do it for them. Providers often have short-term logs of the relevant metadata for security, troubleshooting, and the like; they simply ensure that NSA gets copies as the logs are created (remember: the NSA requires that they do this).
In a phone system, for example, the switching systems that route phone calls log the switching activity. That's metadata. Cell towers log location data (as course as cell handoffs or as fine as GPS coordinates, depending on a host of factors) for service management and troubleshooting purposes. ISPs log requests, including source IPs. And so on. That's all metadata.
The storage requirements for that volume of data are so high that providers typically only retain such data for extremely short periods of time; hours or days at most. That makes them hard to subpoena; so the NSA's PRISM program allows them to simply get real-time copies of those logs, and they handle the data-retention for their purposes.