The “promote the general welfare” clause covers it.

My guess is that it is a composite developed from the GIS databases of the various agencies. If I were doing it, I would send a Freedom of Information request for the data to the agencies and then compile it onto one display map. That would be much faster and easier to do than try to get zip code level data, which was accurate.

The Agencies which manage the lands. Typically, Bureau of Land Management, The Forest Service, and National Park Service allow dispersed camping, unless they have an issue requiring regulation. The typical maximum length of stay is 10, or 14-days.

Well, since the western states, acknowledged Federal ownership of Lands within their state, and agreed to ever challenge that ownership as a condition of statehood, teh western states have no legal standing to challenge.

And, while the Constitution makes no mention of any federal agency managing the lands, it does acknowledge the right of Congress to manage. So, establishing agencies within the executive, and designating management of resources of the United States, to those agencies is entirely within the powers given to Congress.