Who issues the permit depends on where you live. Inside Oxford, the city's Building division issues residential permits and does the inspections, and it enforces the erosion control rules that matter when we open up the ground. Out in the county, Lafayette County requires a building permit for homes in subdivisions, and the application and plans go to its Development Services Department in person or by email.
Whether a garage floor by itself needs a permit isn't something I'll guess at. We check with the office that covers your address before work starts, and when a job needs one, we handle the paperwork and line up the inspections. One soil data site calls Lafayette County's expansive soil risk real but uneven, worst on lots with poor drainage, cut-and-fill grading or old plumbing leaks, which is why we dig out soft spots and settle drainage first.
We're based in Oxford and work across Lafayette County and the cities around it, including Batesville, Holly Springs and Pontotoc. On a lot of garage jobs we find an old slab poured on raked dirt with nowhere for the water to go, and we fix it the same way wherever the house sits.