You should supply the relay with the voltage its coil is rated for. If it's a 5 V relay, then you should supply 5 V. There are certainly plenty of 12 V relays on the market.
If you need 5 V from your 12 V supply, I would recommend using a step down module or a 5 V voltage regulator if you need to get 5 V from your 12 V supply. The former is more efficient, but a little more expensive, but not by much.