This page offers some information on crafting the slingshot.
Slingshot DIY Recipe
To obtain the recipe, you’ll need to buy it from Timmy for 300 Bells at either Nook’s Cranny or Resident Services. There is only one slingshot DIY and there is no Flimsy Slingshot, meaning that the tool will last longer than your other tools if you crafted Flimsy ones.
To craft the slingshot, you’ll need 5 pieces of hardwood. Alternatively, you can buy the same slingshot for 900 bells at Nook’s Cranny.