Mohican Outdoor Center, Blairstown, NJ

They have a nice group camp site with a roof to give you some relief from the rain. There is a great day hike (including a fire tower you can climb) by taking the Rattlesnake Swamp Trail to the Appalachian Trail and then returning by the Rattlesnake Swamp Trail. They also have canoes and kayaks you can paddle on the pond.Phone number: 908-362-5670
Distance from Midtown: ~80 miles ~2 hours 
Reservations required: Yes, and someone in your group needs to belong to the Appalachian Mountain Club
Type of outing: Camping 
Age recommended for: Middle School, High School
Level: 4 
Recommended Season: Spring-Fall
Driving Directions: See their web site at http://www.mohicanoutdoorcenter.com/.

Fahnestock State Park

They have four group camp sites which they rent at $1/person/night (minimum 14 people). Facilities include an outhouse, a fire pit with a grill, and a water spigot near the parking area. 

Phone number: 845-225-7207
Distance from Midtown: ~80 miles/2 hours
Reservations Required: Yes
Type of Outing: Camping
Age recommended for: Middle School, High School
Level (1-5): 3
Recommended Season: Spring-Fall
Directions: North on Taconic Pkwy to west on Route 301 to south on Dennytown Road