controlling one 110v device from three inputs
Easiest way is to use 3 opto  isolators with open collector as "Wired OR" logic and hold cap for a few cycles of half-rectified input current to drive one DC  relay ... or then drive an opto triac or drive AC relay depending on rating.

DIP-8 IC Sockets in veroboard if you can choose proper insulated wiring.
Is any DC available?

Does that seem doable?

Do you want off-the shelf- DIY or cheap or small?
How about Triac Wired OR?