I'm no expert, but when I was educating myself about building a bog, I read alot that people use pvc tubing. They put it down with a few arms going in different directions, and drill holes in it, and maybe use some screen around the holes. It rests towards the bottom of the bog (you have to put it in before all the soil). Then it sticks out a little at the top, and you can put an adapter on the top, so you can attach a hose to it, and water the bog that way. You do it very slowly.
Do an internet search on bogs, or look over old posts here, and I'm sure you can find articles about it. I'm thinking you could water it from the top, VERY slowly. Good luck!
P.S. Did you line yours, or did you not need to? You need to keep it moist all summer, otherwise it is more like a vernal pool...which dries up in the summer. Bog plants would probably not survive that dryness.

I thought about the pvc pipe idea. With all the rain we've been having, I can't imagine the bog ever drying out.

If you think it will stay moist all summer, with just some extra watering, then I'd say go ahead, and just deal with it that way. You didn't mention how big your area is.

This area is about 20' x 30' with no liner. Stays pretty moist. I have Jack in the Pulpit, trillium, ferns and other moisture loving plants throughout my back woods. I can similate rain with overhead sprinklers that are positioned off the vegetable garden adjacent to this boggy area. I think the wet spring into summer we are experiencing in the NE will buy me a season before I have to figure out a pvc pipe with perforations and screening next year. I can feed the pipe off the well we use to irrigate the vegetable garden.

That's a big area. I'm afraid you've expanded beyond my knowledge and experience base! Your woods sounds lovely. I have woods too, here in Indiana, but I only have dogtooth violets and lots and lots of invasives. If I were you, I'd take a little longer to be sure to set it up right. Sounds like you have a fair amount of moisture to buy you some time. Good luck!