The reason why something like low lvl PvP does not exist is that low lvl PvP chars or pretty much any low lvl char is not killing machine. In fact they are dieing machines. Also level difference between players makes everything even more messy than it already is and when you PvP at mid level then you can always expect that higher or max lvl player will come around and you are pretty much screwed. But thats also shame because something like low PvP would be certainly fun with worse gear etc. But I can't imagine how to implement such thing when we can't see our opponents level. Anyway things like PvP grounds could represent some kind of an exp gain for PvP chars. You join few fights in such PvP arena and reward would depend on your level scale. Lets say that players from 6 to 11 lvl would be able to join only something like "training ground" with no real reward from winning but some nice exp. From 12 to 17 lvl they would be able to connect to certain PvP arenas. At these mid levels they could be rewarded only with caps and/or exp (maybe possibility to choose between these or combine them). At higher levels 18-21 they would get also items as reward or maybe something like title to their name (when it comes down to tournaments then maybe cars as reward etc). Anyway PvP statistics should be implemented to "Kills" tab in pipboy. Number of killed players, deaths (not from environment), "honorable" kills (kills of players at your or higher lvl) etc. So when number of you "kills" exceeds number of "honorable kills" like 8 times you will get something like "Chicken" or "Psychopath" as prefix to your name (or just like player flag meaning we get reward for killing such guy or something). Such features could make competing in PvP even more interesting.
Anyway leveling up PvP char in regular world using some "natural" means is tricky. Even when you find something that could work like source of exp its always hard to balance it and there is always abuse issue. But PvP/training grounds that stand apart from regular world could be easier to balance and abuse issues would be also not such threat there. Anyway if devs would just consider something like alternative means of exping for PvP chars or such training/PvP grounds then it will be more than appreciated.
Copy+paste from another topic. That idea about exping through PvP grounds was pretty much just a long shot but some particular arenas could certainly serve even this way but thats just like a really last thing I would expect from PvP grounds.