There is a primitive campground at Fort Leonard Wood:
Piney River Camp - DOD
Dispersed sites, No Water, RV Fee: $0, Tent Fee: $0, Reservations not accepted, Military only, Elev: 781', Nearest town: Fort Leonard Wood
Although restricted to use by military personnel only, some bases allow family members to be "sponsored" by a military member. You or you son might check to see if Fort Leonard Wood provides for that.
Nearby Waynesville is home to Roubidoux Springs City CG with:
Total sites: 26, RV sites: 26, Elec sites: 26, Water at site, RV Dump, Showers, Flush Toilets, RV Fee: $20, Tent Fee: $20, Reservations accepted, Open all year, Pay at City Hall, Bathhouse under renovation, Elev: 781', Tel: 573-774-6171, Nearest town: Waynesville
There are several US Forest Service and state campsites in the Fort Leonard Wood area as well:
Mill Creek - USFS
Elev: 753', Nearest town: Newburg
Lane Spring - USFS
Total sites: 18, RV sites: 18, Elec sites: 6, Central Water, No RV Dump, No Showers, Vault Toilets, Max Length: 34', RV Fee: $8-15, Tent Fee: $8, Open Apr-Oct, Elev: 863', Nearest town: Vida
Big Piney Trail Camp - USFS
Total sites: 2, RV sites: 2, Elec sites: 0, No Water, Vault Toilets, RV Fee: $0, Tent Fee: $0, Elev: 1130', Nearest town: Licking
Paddy Creek - USFS
Total sites: 23, RV sites: 23, Elec sites: 0, No Water, Vault Toilets, Max Length: 34', RV Fee: $0, Tent Fee: $0, Donations accepted, Elev: 890', Nearest town: Licking
Richard F. Clement Memorial Forest - MDC
No Water, No Toilets, RV Fee: $0, Tent Fee: $0, Elev: 1165', Tel: 573-729-3182, Nearest town: Salem
Schlicht Springs Access - MDC
No Water, Vault Toilets, RV Fee: $0, Tent Fee: $0, Open all year, Elev: 810', Tel: 573-368-2225, Nearest town: Crocker
I'm sure there are commercial campgrounds in the area as well. And don't forget to check out Historic Route 66 which runs through that area.
WxToad
2005 Lazy Daze 26.5 Rear Bath
NE13