I've never built a server and never migrated a server. Is this way out of my league? To answer this if you have any other questions I'll try to respond to the best of my ability.

A server, potentially a simple file server, is not necessarily out of your league. But if all they need is simple file sharing, you can probably get your bang from your buck with a NAS.

Any low end Dell or HP would be fine. Don't "build" servers, it's a horrible practice in general.

Will there be any benefits to just getting them an NAS instead of using a full on server with a server OS?

Yes, simpler and less maintenance if all they need is file sharing, plus more features.

The current server's OS is Microsoft media home. What type of migration software would I use if I were to switch to an actual server OS / NAS?

If you just deploy a NAS, then it's typically a matter of just copying files, setting permissions on the folders, and creating user accounts for people.

I'll be honest even this seems a bit out of your league, but deploying a simple little NAS should be fine.