Not just your stomach, but your entire digestive system actually. It’s called the enteric nervous system, and it’s absolutely fascinating (the disfunction of which has recently been linked to a lot of different diseases and disorders). From Wikipedia:

The enteric nervous system (ENS) is one of the three divisions of the autonomic nervous system (ANS), the others being the sympathetic nervous system (SNS) and parasympathetic nervous system (PSNS). The ENS is formed from the myenteric plexus, and the submucosal plexus, and consists of a mesh-like system of neurons that governs the functions of the gastrointestinal tract, including motility and secretion, and is known as the “second brain”.

The enteric nervous system is capable of operating independently of the brain and spinal cord

The enteric nervous system in humans consists of some 500 million neurons, 0.5% of the number of neurons in the brain, five times as many as the one hundred million neurons in the human spinal cord, and about ⅔ as many as in the whole nervous system of a cat. The enteric nervous system is embedded in the lining of the gastrointestinal system, beginning in the esophagus and extending down to the anus.

The stomach and heart both have neurons in them.

Only in the sense that the gut is deeply tied to your emotional state. Example: “that sinking feeling in the pit of your stomach”

Your stomach hires your brain to think for it.