For example, if I wanted to describe something that had just happened but also happened to me on a recent camping trip. Would it be grammatically correct to say, "that happened to me on camping too"?
I would use "while camping" here, e.g. "that happened to me while camping too".
Sometimes people will say something happened "on camp", or "at camp". To take an slightly dated pop culture example, one of the characters in the movie American Pie often tells anecdotes that start "this one time, at band camp, ...".