Yes a Ford F150 truck can and should easily tow a camper and even a camper up to around 13,000 lbs with ease.
A Ford F150 truck has the ability to tow up to 13,200 but that is for a regular cab, long bed 2-wheel drive, 3.5-liter Ecoboost with the maximum tow or heavy-duty payload.
I would recommend for the larger campers to have a transmission cooler installed on the truck to keep the transmission fluid and transmission cool especially when towing up hills a lot.
Most Ford F150 trucks even the older ones can easily tow a camper around with ease and should have no problem towing with it.