Chrysler minivans are - or at least used to be - wonderful family cars. I loved ours (we had an '88 Dodge one, sort of the forerunner to yours). We pulled a popup (yes, with tented sides) with it, and did fine until we took it to the Black Hills one time. The minivan had trouble pulling the popup up the long climb into the Hills - we had to stop several times to let the engine cool down. Mostly we stayed in flat country, and enjoyed our camping trips, but we knew we wanted to go to the mountains more, so we traded the minivan in on an SUV, which could handle the popup. THen that SUV (2002 Ford Explorer) had trouble with the first full-height trailer we bought, a hybrid. So we traded that in for a truck - and you can see the result in my sig.
Good luck to you - but I suspect you will not be happy with a full-height trailer and that minivan, unless you camp in only flat country, and don't drive into a headwind.
There are hard-sided popup campers, but they might be tight for a family of 4. Still, you might check out A-liners and similar.
