I am trying to preserve home-made mayonnaise so that it can be shipped to family and friends, without the hassle of putting it in dry ice packs or the like. Is there a way to preserve it and ship the bottled mayonnaise by regular mail or courier?
Nope. The only way you can really preserve things at home for room temperature storage is by canning in a boiling water bath or pressure cooker, and the heat from that will break the emulsion of your mayonnaise, completely ruining it. On top of this, mayonnaise will also tend to break at room temperature, so it won't work even if you make it safe.
Commercial mayonnaise uses pasteurized ingredients (in particular the eggs), and presumably the manufacturing process takes additional steps to avoid recontamination before it gets sealed up in jars. It also has additional stabilizers to keep it from breaking while sitting around at room temperature. The result is apparently shelf-stable, but even if you add emulsifiers/stabilizers to yours and get it to not break during shipping, I would be really hesitant to assume you've gotten it as bacteria-proof as the commercial stuff.