Insulin is a protein and like pretty much all proteins, there's a gene for that.
It is usually called the INS gene or insulin gene, but has also been called the MODY 10 gene, the proinsulin gene, and a few other names, and it's located on the short arm of chromosome 11.
Mutations in the INS gene are a rare cause of diabetes; mutations in both copies of the gene lead to neonatal diabetes, and mutations in just one copy leads to MODY 10. Some mutations lead to an even rarer condition where the body makes too much insulin.
The gene actually does not code for insulin itself, but for proinsulin, which is insulin attached to c-peptide.
At 1,430 base pairs, the insulin gene is among the shortest human genes, which is probably why mutations in this gene are uncommon.