I purchased a Fifa 16 key from an online seller, but haven't got a copy of Fifa 16.
Is there any way to download Fifa 16 from origin without paying for the key, again?
Assuming by "Origin" you mean the EA digital distribution client, you can just right-click the icon in your task tray, select "redeem product code" and then activate the key in there.
If you don't have the client, you can download it from https://www.origin.com/en-us/download. You'll need a free EA account to use it.