4-pin trailer connector just has wires for lamps, and aren't normally that high current.
I'm assuming the 12V accessory plug you are talking about is the cigarette lighter? It may provide enough juice for the jack, but you'd have to find out how much power the one in your vehicle can put out, and compare it with how much power the electric jack needs.
A small battery on the trailer would do ok, but it may be a hassle to keep it charged (as again, the 4 way doesn't have an always-on power supply for charging the battery, even if you use the trailer all the time, so you should put a trickle charger on the battery).
If this is something you want to do, I would suggest seeing if upgrading the plug setup on your vehicle & trailer to a 7-way, IF it can provide the power needed to run the electric jack, or if you use the trailer pretty regularly and you install a battery on the trailer for the jack.
Or, you could run wires from the vehicles battery, with a fuse, to a second plug (just a 2-way, power & ground), that you mount at the rear of the vehicle, and use that plug to power the electric jack.
Electric tongue jacks are common on travel trailers which have a hot lead in the 7-pole connector for charging the trailer battery and running 12 volt appliances while travelling. Even with two large deep cycle batteries on the tongue to power the jack, it always runs better when the trailer is connected to the truck's system with the engine running.
It's been a long time (27 years), but I think that there is a 30 amp fuse on the hot line in my old van. The hot line is 10 or 12 gauge. Electric jacks can draw a lot of power if the load is heavy.
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