The odor comes about due to bacterial growth in stagnant water with lack of air. It can be expected in water systems that are not used for some time, especially if the water put in them is NOT from a chlorinated city water system, as campgrounds tend to be. Natural well or spring water, for example, is especially prone to this.
The solution is, firstly, to completely drain your water system, especially the water heater, between trips. In addition, sanitizing the system at least once a year using chlorine bleach is also helpful. For what it is worth, even though the water smells bad when you shower or flush the commode, it won't hurt anything. Tastes awful, too, if anyone holds their nose long enough to drink it. But I'll point out that a lot of famous mineral waters have the same odor for the similar reasons. I recall a trip to Bath, in the UK a few years ago, where the curative properties of the mineral water were legendary (and IMHO mostly trumped up hokum), and that water reeked of rotten eggs.
I think I'd stay away from running vinegar through the system, though, because that would be corrosive to any metal parts, such as the tank of the water heater.