NO national park or national monument anywhere will allow collection. Very, very few State parks will allow it, either. (See caveat below.)
However, National Forests do give out collecting permits (for a fee) as do many State forests -- except for areas of those forests designated as "wilderness". Check the U.S. Forest Service website or visit one of the local offices of whatever National Forest is closest to you. Ditto state forests.
Finding decent places to collect, however, is a different question. You find very few truly collectible trees IN the deep forest or in open fields. Collectible areas are often quite dangerous -- cliffs, bald mountaintops, and stream/river edges, where for one reason or another trees struggle to survive. There are very few of these in the eastern USA that aren't in national or state parks or wilderness areas.
Many of the best collecting sites in the east will be in and on private lands, but these again can be dangerous -- Quarries, mine tailings, and reclamation areas, barren, logged-out lands, etc. Getting corporate permission is often difficult to impossible; corporate lawyers always are concerned about liability and damages.
The best collecting in the east is, I'm afraid, in urban and urbanizing areas, urban renewal sites, local park renovations, etc.
The west is different, but there are all sorts of environmental issues out west that we don't have in the east (not that many western states seem to give a damn).
Caveat regarding collecting in parks: Occasionally (rarely) a park will have a day for clearing invasive species out of native environments. A few of the invasives they'll go after (Chinese privet, for example, or Brazilian pepper in Florida) might make decent bonsai.