If you want an authoritative click you'll need a contactor
If you get one surplus they are pretty cheap. It's just a relay writ large.
You may have to get a much heftier transistor to activate one. If you choose one of these you will need a 24 or 28VDC power supply. If you look around you might find one that runs on a lower voltage but I've not seen one that runs on 5v.
Be sure it's a DC coil one, there are ones that have a coil that needs AC.
Your next project could be a motorized switch, they look so rube goldberg!