I did a long distance drive about ten years ago via the I-40, and decided to check out the Canyon so long as I was passing by.
I switched to State Route 64, which took me directly to Grand Canyon National Park Airport (GCN), near the South Rim. There are 2 or 3 companies there selling helicopter flights over the canyon.
There are no special documents or anything required—you pay the money, they'll put you on board.
Here are a couple of operators I found with a quick search:
It looks like flying out of GCN is cheaper than from Las Vegas, probably because you're already at the Grand Canyon (that is, they don't need to fly you there and back again).
My experience: it's a great way to see the Canyon in a short amount of time. The views were incredible, and I was able to get a real feel for just how immense it is.