You need no permission from any authority to land on the moon. It is a celestal body, and as such cannot be claimed by any nation. The only authorisation one would need is to use the airspace of the country in which the launch will take place, in this case, is it in Kazakhstan. I would suspect that State Department permission has been granted for the use of the equipment?
State would need to give permission to export some of the equipment to Kazachstan probably (some components might fall under the export restrictions on missile technology).
There are treaties in place on the use of space, but those only apply to earth orbit (mainly geostationary orbit) where space is limited.
There are also treaties banning commercial use of the moon, but that is mainly restricted to mining. The moon has similar status to Antarctica.
A 1967 treaty made the Moon the property of all the world's nations.
This might, just might, be a first step to new explorations.
But launching a small probe on a commercial satellite launch vehicle is a very long way from Moonbases, nothing will happen until a fully reusable and relatively cheap launch vehicle to low Earth orbit is developed.
In addition, the Trailblazer mission should provide the opportunity to photograph the equipment left behind by past Apollo and Russian landings putting an end to suspicions that the Moon landings were faked.
I agree about the main thing stopping this from happening is our current unavailability of a cheap spaceplane to take us into low Earth Orbit. We need to replace the shuttle in the next 5-10 years or so(by then it will be too old). Instead of flying Earth-Moon direct as with Apollo a better way to do it would be to take off in a spaceplane, dock with the International Space Station and then enter a separate spacecraft to take you to the moon. This would be much more efficient and probably cheaper than building a massive rocket like the Saturn V. But first we need to get the ISS completed and an efficient spaceplane of some sort up and running.