The question isn't "can you", but "should you".

An engine that's always working at near it's maximum capacity will fail long before an engine that's working at a quarter of it's capacity. Most people wouldn't dream of constantly running their engine bouncing off the red line of rev limiter. The same applies to towing; if you frequently tow large, heavy loads (for instance, earth moving equipment), you want to get a vehicle that's rated for much higher tonnage than the weight you'll regularly be towing. Given that campers are usually very light weight (but only slightly more aerodynamic than a brick), you can get away with towing on in a car infrequently. You should probably not do it daily.

You may also find that it's less fuel efficient to tow a heavy load in a small-ish car than the same load in a light truck.

(BTW - I'm generally opposed to taking vacations in this way. I prefer my vacations on a motorcycle, or on foot with a backpack. I'm not currently in the kind of shape I would need to be in in order to do bicycle camping.)

Edit: I don't have a truck. It's cheaper for me to rent one on the rare occasions that I need one than it is to buy one and deal with the associated costs of owenership. That said, the Home Depot rental trucks suck, because they're solely RWD, and they have no grip on my road unless there's a literal ton of weight in the back.