"What she did have were Bertie Bott's Every Flavor Beans, Drooble's Best Blowing Gum, Chocolate Frogs, Pumpkin Pasties, Cauldron Cakes, Licorice Wands, and a number of other strange things Harry had never seen in his life" (Sorcerer's Stone 101).
Unfortunately, a major ingredient when making chewing gum is gum base. Gum base is not cheap. And since this is probably the only time I will ever make gum, I will not be purchasing gum paste, which seems to only be available in large quantities. Also required is citric acid, which is also difficult to find for sale.
So if you would like to make your own bubble gum, a pretty simple recipe can be found here (where you can also purchase the specialty ingredients required), but I will not be making any bubble gum myself.
I suggest that you go to the check-out aisle of your favorite market and buy some.
...at least that's what I'd do. Sorry to disappoint!