Camping is allowed anywhere in state forests, including the Catskills, and Allegany State Park.  No permit is needed.  Read more about specific regulations on the NYS DEC’s web page:

Most state parks require you to stay in their designated campground, unless there’s a designated campsite or lean-to.  Check the state park’s website or search the NYS DEC “Places To Go” web page:

In addition, there are many hotels/motels near the trail.  Using Google Maps is generally the best method to locate them.

AP: Access Point
MFT: Miles From Trail
Type: Bivouac = Tent sites, PubCamp = Public Campground, PrivCamp = Private Campground