A synonym for a peach seed is pit. To feel something in the pit of one's stomach means to feel a sick feeling in your stomach, usually from anxiety. The person in this joke swallowed a peach seed and felt anxiety. This joke is playing on the word "pit" because it's a synonym for seed, and it's referring to someone's anxiety felt in the stomach after swallowing the seed.
She swallowed a peach seed and could feel it in the pit of her stomach .