I control a smoke machine for my halloween display, I use a small relay which can flip an AC circuit, but I'm not using an arduino to power it (it's powered off a 12v relay board connected to a PC. If you're fog machine is like all the ones I've seen, it's simply running the 120v line voltage to the remote and back if you press the switch. It's not a lower trigger voltage to control a relay in the machine itself. Because it's controlling a motor/pump I'd get a more hefty relay that can handle more current than the one you picked out to handle spikes if they occur.
Hey well today I did find that infact the line voltage (240v here) runs straight through the remote, however, the element is heated from the switch on the back so no high current. The pump the remote powers is 240v 16w.