Is it customary to use a transistor at the end of a string of logic gates when powering something such as a relay?
This is a good learning opportunity.
The ability of a logic gate to sink current will be listed as its \$I_{OL}\$ in the data sheet. So, pick the specific OR gate that you want to use and check its data sheet. Keep in mind that the specified current value will not be for an output of 0.00 V, but probably for the particular gates specified output voltage for a low logic level \$V_{OL}\$.
If the current required by your relay exceeds the current available from the logic gate then you must add something to drive the higher current. That added &quot;something&quot; can be a transistor, either a BJT or MOSFET, but the details will depend on the operating voltage of your logic and relay. You have not given us either of those specifications.