My local computer shop installed Windows 7 Home Premium with their OEM key, and I cannot afford to upgrade to a higher version. I have 32GB of RAM but Home Premium has capped the RAM to 16GB, rendering half of my RAM unusable. Is there any way I can change a registry key or something to allow myself to remove the 16GB cap?
These are your only options for Windows to be able to use all 32 GB of RAM.
If you do upgrade make sure you get 64 bit. You say you can't afford to upgrade, then I don't know why you bought 32 GB of RAM for a home version of Windows. If you can afford the RAM, you can afford the upgrade.
Unless they gave you that OEM key itself (the sticker with the license number) they sold you a pirated copy of Windows.
There is not any way to circumvent the 16 GB cap, unless you go even further into software piracy territory.
Or install Linux. It's free and doesn't have these silly restrictions.