It may or may not work, depending of the specific model of the relay.
If there is no datasheet for the relay in question that say about minimum required voltage for the relay to close, then you have to test each relay manually.
Most 6 V relays will work on 5 V. However you may not get exactly 5 V on a 5 V power supply, so it may be a matter of testing it.
All the relays that I have seen need a lot more voltage to turn on than to keep them on. A 6 V relay might turn on at 4.5 V but then the voltage would have to be reduced to maybe 1 - 1.5 V before it turns off.
It probably will work but it's not something to do if you are producing more than one circuit (for production).
For that you could add a bootstrap circuit to momentarily apply 10V to the relay to pull it in (but that requires a capacitor, two diodes, and two transistors, example circuit below).
It may or may not work, depending of the specific model of the relay.
If there is no datasheet for the relay in question that say about minimum required voltage for the relay to close, then you have to test each relay manually.
Most 6 V relays will work on 5 V. However you may not get exactly 5 V on a 5 V power supply, so it may be a matter of testing it.
All the relays that I have seen need a lot more voltage to turn on than to keep them on. A 6 V relay might turn on at 4.5 V but then the voltage would have to be reduced to maybe 1 - 1.5 V before it turns off.
I would expect that to be more of a problem at high temperature, when the relay coil resistance rises.
Remember that magnetic strengths is as function of current, not voltage. The voltage is just whatever is required to push the needed current through the resistance.