I've hunted all of my life right here in Virginia, I would like to go on a Elk hunt in the coming years, I can't imagine the excitement

What state closest to Virginia has a good Elk population?

Thats a good question. I have not hunted that area at all. I don't know of anything closer than Wyoming

Wyoming or colorado...unless you want to try for one of the impossible Pa, Oklahoma, arkansas, or Kentucky tags...

Try western Pa there elk herd is making a come back and they have been allowing people to hunt there for the past two years.

the virginia dept of wildlife says that they have elk that have come in from kentucky, try for one of them. all you need is a deer tag and a deer season( checked into it myself.) may be few and far between, but someone has to shoot them, i guess.

Getting a tag in PA is extremely hard. Less than 100 given out. We only have a heard of 700 or so. I'd save up and go out west. But it couldn't hurt to try a PA elk hunt

I'd worry less about the distance from Virginia (you're probably planning on driving) and concentrate on the areas where you're most likely to draw a tag. Depending on how soon you want to do this you'll be presented with a great number of choices of State, GMU, time of year, etc. Costs of the trip are also a big factor, of course.
If you're patient and you really want a special hunt, you may want to apply for one of the limited access areas offered by several western States. You won't draw a tag for a while but will collect preference points which will eventually lead to a tag. This isn't the best solution for the eager hunter but its a strategy which is most likely to bring success when you do finally get to go there.
I have a friend who applied to hunt a very limited area in New Mexico for years and finally drew a tag. The result was a massive 6x6 bull. The rack doesn't fit in any room in his home so it sits out on the back porch.
Finally, cow tags are usually easy and far less expensive.

My neighbor in utah last year shot a 392 3/8 elk in one of the limited entry units. I have seen 410"+ bulls in limited areas while deer hunting.

PJ -
I used to own "Virginia Outdoorsman" and heard all kinds of stories in VA but a couple of years ago some character shot what he thought was a "large deer" in South West VA ..drove around showing it off and finally was stopped by Fish & Game..It was a small cow elk...
Sad part is there was NO LAW that stated you could not shoot an Elk in VA, so he received no fine..Elk migrated from Kentucky
Maybe you will not have to go far at all!