You need to apply to the Isle of Man police for a "Regulated Weapon Certificate". This covers air weapons up to 6/12 ft lb (for pistols and long rifles respectively, as in Great Britain), as well as shotguns and crossbows.
The Regulated Weapon Certificate is broadly similar to a GB shotgun certificate. It permits you to acquire and possess as many regulated weapons as you choose, and has a 10-year duration. You must keep self-certified records of regulated weapons that you acquire and possess. You must also provide secure storage for regulated weapons AND their ammunition - including of course air rifle pellets. For all practical purposes, this means proper gun/ammunition safes. You will also find, if you go to buy pellets from a gun shop on the Island, that you'll need to produce your certificate in order to do so.
I hope that helps.
Adrian