WikiTravel doesn't have an article on the Gates of Hell, but WikiVoyage does.
There's at least four tour companies listed. Two international and two local.
You apparently can drive there but it requires a 4x4 and you'll be driving across the sand rather than on a road. They suggest that you won't be able to get there in a rental car but don't give any details about renting a 4x4. But to sum up:
For those who are more a bit more adventurous or daring, it is possible to arrange a Derweze trip oneself, but it takes considerable work to make sure you end up with a safe trip. It probably helps to have Turkmen contacts who can point you to someone with the correct type of vehicle.
There's a ton of extra information in the WikiVoage article I could quote here, or you could just pull your finger out and read it all yourself (-;
Promoted my comment to an answer under threat.