Combine the ham and pickled peppers in a food processor and pulse until they are finely chopped and well blended, scraping down the sides as needed. Avoid overprocessing; the mixture should have a bit of a coarse texture, not become a smooth puree.
Add the mayonnaise, vinegar, and mustard to the processor and pulse three or four times to blend. Taste the mixture for seasoning, adding a bit more vinegar or mustard to taste. If the mixture is a bit dry, add another tablespoon or so of mayonnaise for a more spreadable texture.
Transfer the deviled ham to a serving bowl. If you are not serving right away, cover and refrigerate until you are ready to serve. The flavor will be best if made at least 2 hours in advance, but it can be prepared up to 3 days ahead. Allow the spread to come to room temperature before serving.