There is no public transportation other than official vehicles on base. It is not uncommon for certain vehicles to be assigned to higher ranking individuals and also vehicles for groups of people. Access to a vehicle is not a problem. However, as there is only 1 road once you leave the base, to go anywhere else requires a 4 wheel drive vehicle. There are some basic maps to show you where some of the "tourist" sites are, and some areas that are cleared of the bigger rocks to get to other areas that are no longer used. But basically, you point the car in the direction you want to go, and go that way. There are no flashing lights, no road markings (with one notable caveat) and no local police to tell you how to drive.
There ARE posts with reflectors between the main base and BMEWS (Ballistic Missile Early Warning System) so that in the darkness and in blizzards, you can still know which direction to go.
Make sure that you always travel with a companion (see my warnings and dangers) and have fully fueled your vehicle before you go on any adventures. There is only one fueling station for hundreds of miles of grueling tundra. I don't think you'd get to the next one on a single tank of fuel!
Getting to Thule is not an easy feat. Thule airbase is, of course, an Airforce base. I am not sure about the Canadian service there, but I would guess that you cannot get to Thule from Canada unless you have orders to it. (or you want to cross the icepack in the winter ) Getting there from the U.S. is likewise difficult. Unless you have travel orders cleared by the Air Force, you cannot fly on any of the Military flights there. And no civililan flights originating in the U.S. go there.
Getting there from Denmark is a different story. I don't know the entire procedure, but I know that there is a weekly flight into Thule from Denmark (Copenhagen I believe) and in the summer, there is normally more than one. I know it has to be cleared through their embassy, but I do not believe it is nearly as difficult to get there from Europe as it is from the Americas.
There are a couple of ships a year that port in Thule, but those are fuel and supply ships from the U.S., Canada and Denmark. I don't know if it is possible to book passage on those ships at all. As they are merchant ships, and money is the international mercantile language, I would imagine the right contacts and the correct denominations could get you a ride.