What may help your research is knowing that percussion caps for firearms predates cartridges, which I think is what you mean by bullets. A bullet is a projectile, a cartridge is a complete package of bullet, powder, and primer in a casing. People often call the whole thing a bullet, but that's not quite right.

Prior to the invention of cartridges, firearms were exclusively muzzle-loading types where the powder and a cloth were tamped down the barrel and a ball (bullet) was dropped in and tamped afterwards. A cap was fitted on the "lock", the part of the gun that houses the trigger and hammer. The hammer struck the cap, igniting the powder and propelling the ball.

The first caps were invented at beginning of the 19th century. Prior to this, the powder was ignited by flint striking steel to make spark. That's where we get the term "flintlock rifle".

But to answer your question, I'm afraid I don't know off hand or have any scholarly articles that I can point to that tell us who invented the first one.

Mods: I hope this is an acceptable response! I'm going off my own knowledge from my own study of firearms and a lifetime of shooting sports. I think most of what I shared above is somewhat common knowledge among shooters.