You're going to be hard pressed to find a campground closer to Harrison Township than Algonac State Park.
There is the St. Clair RV Resort
- which is 29 miles straight up I94 from Harrison Township that while a little farther - is probably an easier drive than the trip to Algonac State Park. It's also got some amenities that the state part doesn't offer - like full hook-ups, pool, etc....
Aside from that - your next closest set of options would be the parks along the I75 corridor in north Oakland County. These include Holly Recreation Area (part of the state park system) or the Oakland County Park system
(Addison Oaks, Independence Oaks, Groveland Oaks) ... however, the closest of these would be Addison Oaks which is roughly 30 miles of "non-freeway" from Harrison Township. Note that Holly Recreation Area and the Oakland County parks do not offer "full hookups". Sites are "electric only" with a central dump station.
There are a couple of "private" campgrounds tucked in and around the north Oakland county parks - which you find by looking on Allstays. However I've had zero experience with any of them - so can't provide much in terms of recommendation. Of the parks along the I75 corridor - Holly Recreation Area and/or Addison Oaks would be my preference.