In a large bowl, mix chicken, lettuce, ham, provolone, pepperoni, soppressata, banana peppers, and onion.
In a medium bowl, stir mayonnaise, vinegar, and Italian seasoning. Pour over chicken salad and toss to combine.
Scoop 1 heaping cup chicken salad onto the center of each tortilla. Fold in left and right sides of tortilla, then fold end closest to you over filling. Tightly roll up bottom end, keeping sides tucked in, to form a burrito shape. Cut each wrap in half.