Rifles and Shotguns can be mailed via USPS by a nonlicensee.
Any other firearm, such as handguns, frames, receivers, AR lowers, pistol grip firearms that expel a shotgun shell and NFA firearms are mailable only by licensed dealers and manufacturers.
Having an FFL doesn't mean FedEx and UPS will ship. You have to be an FFL enrolled in their FFL shipping program.