Lots of things you wouldn't expect are made from pigs but bubblegum appears to be one of the few things that isn't at least not nowadays.
Elastomers: provide the elasticity or bounce, and can be natural latexes (e.g. couma macrocarpa (also called leche caspi or sorva), loquat (also called nispero), tunu, jelutong, or chicle which is still commercially produced), or synthetic rubbers (e.g. styrene-butadiene rubber, butyl rubber, polyisobutylene).
Resins: provide a cohesive body or strength, and are most often glycerol esters of gum, terpene resins, and/or polyvinyl acetate.
Waxes: act as softening agents and are most usually paraffin or microcrystalline wax.
Fats: behave as plasticizers and mainly come from hydrogenated vegetable oils.
Emulsifiers: help to hydrate, the most common being lecithin or glycerol monostearate.
Fillers: impart texture and the most commonly used are calcium carbonate or talc.
Antioxidants: protect from oxidation and extend shelf-life; the most common type is BHT.
Sugars: Either natural or synthetic usually dextrose.
Glucose syrup: Another sweetner normally tapioca syrup
Flavourings: Could be anything but unlikely to be a natural product.
Colouring: You really don't want to know