I've had plugs as recently as 10 years ago last the life of the tire. I don't think plugs are inherently more prone to failure than a patch, but a patch is certainly a better way of doing it since it covers a larger area - less pressure on the repair.
Discount Tire (if there is one near you) will patch a tire at no cost, provided the tire can be patched - the hole has to be a certain distance away from the sidewall, etc.
You do not have to purchase the tire from them for them to patch it. I'd imagine that with a trailer tire, you'd ideally want to take the wheel off the trailer and bring it to the store in your pickup or trunk.
They will also rotate your tires at no cost.