What's the purpose of sugar in a pickling solution?
The sugar is mostly just for flavor.  I use sugar when making pickled beets and eggs, but don't use it in my dill pickles or pickled peppers and onions.  It just depends on if you are trying for a sour, sweet and sour, or sweet pickle (note that there is no vinegar in many fruit pickles).

Before you decide to run off and leave out the vinegar, however, I would note that for pickling safety purposes there has to be enough acid to keep botulism causing bacteria from forming and peppers are a low acid vegetable.  If you wanted to eliminate the vinegar, you'd have to move to pressure canning.