Hours, phone numbers, and directions available from the U.S. Postal Service website.
For quick searching, note the zip codes below.

The largest post offices are in Milford PA, East Stroudsburg PA, and Stroudsburg PA.
Drop boxes for other carriers (e.g., UPS, Fedex) are nearby.
Smaller post offices may not be able to guarantee overnight service.

PENNSYLVANIA
North end of park
200 West Harford St., Milford PA 18337
103 Doodle Hollow Rd., Dingmans Ferry PA 18328
(Rt. 739 west from Rt. 209 1.2 miles, turn left onto Silver Lake Rd., continue 1 mile to Doodle Hollow Rd. on the left)
Middle of park
911 Bushkill Falls Rd., Bushkill PA 18324
(Turn west onto Bushkill Falls Rd. at the blinking light in Bushkill)
9 South Dartmouth Drive, Marshalls Creek PA 18335
South end of park
Shawnee Square, Buttermilk Falls Rd., Shawnee PA 18356
(In a small shopping mall south of the village of Shawnee, at the top of the hill)
"0" River Road, Minisink Hills PA 18341
200 South Courtland St., East Stroudsburg PA 18301
(This street is one-way northbound)
12 Main St., Delaware Water Gap PA 18327
(Entrance is on Shepard St.)
701 Ann St., Stroudsburg PA 18360
(Large facility one block south of Main St. at 7th St./Park Ave./Rt. 611)
403 Delaware Ave., Portland PA 18351

NEW JERSEY
North end of park
264 Clove Road, Montague NJ 07827
(Off Route 206 just outside the park boundary)
Middle of park
119 County Road 560, Layton NJ 07851
(South side of Rt. 560 just east of the curve at the light in Layton NJ)
South end of park
7 Decatur Avenue, Columbia NJ 07832

The "post offices" in the General Store at Millbrook Village NJ and at Walpack Center NJ are both historic structures; they no longer function as actual post offices.