Different airports have different security classifications depending upon the nature of their operatons. In the case of MLB, that airport handles both scheduled commercial passenger airlines [both domestic and international flights] as well as GA operaitons. Its security restrictions are pretty standard for an airport of its classification. You are not allowed to carry a loaded firearm into the Aircraft Operations Area, that is the area that is accessible for aircraft [hangers, parking ramps, etc.], in either the Secure IDentification Area [the AOA around the passenger terminals] or the less restrictive AOAs located at the Fixed Base Operations in the General Aviation sections of the airport. This is a federal mandate. The areas with unrestricted public access, outside of the designated passenger terminal areas, are not subject to these restrictions and concealed carry, with a CWL, should be legal. Before carrying into any building that houses flight operations, you should check to see what the airport designation for that building is. In the case of most FBOs, the management can provide you with that information. Some, which serve as a passenger boarding area for charter airlines, are now considered a "terminal".
In the case of most GA only airports, those having no regularly scheduled commercial passenger flights or not hosting any certified airline operations [these can include charter airlines whose flights originate or terminate at that facility] are usually free of the more rigourous federal security requirements, though most are posted against trespass in the AOA.
Flight schools could pose a problem under state law. In the case of a publicly owned airport, I would think that local ordinances prohibiting weapons would have no more legal weight than would those applying to parks.
I know this is complicated, but I hope it helps.