According to Wikipedia, slingshots did not exist prior to the invention of vulcanized rubber.
Slingshots depend on strong elastic materials, typically vulcanized natural rubber or the equivalent, and thus date no earlier than the invention of vulcanized rubber by Charles Goodyear in 1839 (patented in 1844). Source
The Bow and Arrow date back 17,000+ years in simple terms these store energy in the bow (wood) and launch a long straight projectile (Arrow) they are relatively simple to make from "Natural" elements in the wild in a survival situation.
The slingshot has an advantage over the bow, as the projectiles are much easier to acquire. Finding round rocks in stream bed, is much easier then the time and skill required to make a straight arrow.
There is also an Sling that uses rotational energy to launch a small round projectile.
The slingshot has a couple advantages over the sling: It requires less skill to use effectively, and it can be an effective stealth weapon in heavy foliage (pull and release as opposed to swinging).
While you can argue that vulcanized rubber is a "Natural" resources, it would be exceedingly difficult to create in survival situation.
Other then vulcanized rubber, what (if any) natural resources could I use to make a slingshot?
I suspect the closest you could get to a hand operated, small projectile, stored energy, survival situation built, weapon is a small cross bow