Get another contractor?
Glenn, you don't say where you are located. I assume you have confirmed wiht local building officials that a permit is required. If not, call and ask. Most are very helpful. Not pulling a required permit is a serious matter. While the required inspections may not catch everything (they rarely do) the fact that the person who pulls the permit is responsible for code compliance means that they are concerned with code compliance. Second, if the contractor is asking you to pull the permit, in GA that would be a likely indication that they do not have the required licensed to construct your carport. Do NOT pull the permit yourself (unless you are qualified). Remember inspections may provide another layer of safety checks.
You should also be aware that often disclosure laws would require that you disclose you had unpermitted work performed on the home and could adversely impact a sale, an home sale inspection or an appraisal.