STOP. Do not connect AC power to any part of your low-voltage system.
What you're doing in the low-voltage regime is perfectly fine. However, you are choosing a relay with a 120 volt AC coil, which means you're going to be taking AC 120V power and delivering it to the coil somehow. There is no safe way to do that.
Use a relay with a low-voltage coil and supply it with low voltage.
Have all AC power contained inside proper "Class 1" wiring methods and enclosures. Never mix AC mains and low voltage. Here's how you cross that threshold:
How to get low-voltage that is "on" when the AC power is
Since you seem to want the relay to actuate when AC power is available, I'm presuming you are having your low-voltage circuit react to something fed by AC, such as a refrigerator light, garage door opener light, socket on a generator, supply from a time-of-use switch, sump pump limit switch, or what have you. How do we get low voltage from that?
- If it's a plug, easy peasy - use a "wall wart" power supply or "lump in cord" power supply. I bet you have a drawer full of them. If you like 5 volts, even easier - any USB power supply.
- If hard-wired, then use a 24V thermostat transformer, or 16-24V doorbell transformer. They mount on a common 99 cent steel junction box knockout or cover, and they are cheap commodities.