My unicycle has a flat tire. The hole is at the very base of the valve, where the valve attaches to the tire. Is there a way to fix this without replacing the entire tire? I was gonna put some Gorilla Glue over the hole and see if that did the trick, but thought I’d better check here first.
Usually, holes at the valve stem are difficult to fix because of the high constraints area it is.
However, finding a replacement tube (not tire 
) is easy and can be done in any proper bike shop.
Just write down the size of your tire or bring the damaged tube with you and you can get a replacement for less than $15 (depending on the size & quality).
Don’t waste your time, just get a new tube.
Indeed, get a new tube (the one you need is probably a common bicycle size), and make sure it gets installed without stress on the valve stem.
Gorilla glue is not going to work here.
If the leak where elsewhere, a tube patching kit with its special cement might work.
If you were desperate and already had another bad tube with an intact value stem area, you might be able to cut out an area around that and patch it into this one using a lot of repair cement and/or the sealant goo used when doing tubeless conversions. But there’s no point in bothering with that when you have tubed setup that uses a common tube size.
If you end up taking the whole unicycle to a bike shop rather than changing it yourself, it’s probably worth checking when you get it back that the wheel is on the right way around, and the bearing caps aren’t over-tightened, since those are the aspects that would differ from what their staff is used to.