What size and type of wire should I use for a 1000 ft run to create an standard outdoor 120 volt outlet?

The part of the question about the dequipment "pulling" 80V doesn't quite fit, what you really need to know is it's utilization of the 120V supply - how many amps does it from the 120V receptacle. If this equipment converts it to 80V internally, that's not really relevant.

Assuming the device uses 3A at 120V, you could use #6 aluminum with less than 5% voltage drop at 1000'. Direct burial cable will probably be your best bet. However voltage drop is load-dependent; if someone plugs in something that draws more power, it would see more voltage drop.

That is an expensive wire purchase but it would be even more for copper. You'll have to plan to splice the aluminum to more normal sized copper at the end. You can do things a little more complicated, like supply a transformer at the receptacle with a 240V, which will allow a smaller wire, or step up to an even higher voltage and back down at the receptacle, but at these distances and loads it's not likely to be worth the trouble.