One problem with this approach is that you need to know about the existence of the software in order to craft a FOIA request such that you can see the code, this can mean sending multiple FOIA requests. Combine that with the delay in fulfilling a request, on the order of months, or sometimes longer, and you could be looking at a year or more before you get the source code you were looking for.
Work done by US government employees as part of their official duties is in general not under copyright within the US.
But that is a much narrower statement.
Most software developed using government funds is not exclusively developed by government employees as part of their official duties. Most Government funding of software development goes to contractors, where that rule does not apply.
The no copyright only applies to the US, not necessarily outside. Outside the US it definitely can have a copyright.
There are also some weird exceptions for government employees, specifically for the u.s. postal service and for certain materials created by NIST.
You can absolutely make a request. If you are interested in the software, go ahead. However, the Freedom of Information Act only applies to records, and software is not always considered a record.
So Freedom of Information Act request will work if they're already predisposed to release the software, it's no guarantee.
My understanding is that that only applies to software written by government employees. Put more rhetorically: if the government pays for Windows (which it does), does that mean you can access the Windows source code via FOIA request?