Are you a member of a club? There are team and individual Postal competitions run by the NSRA, and probably also county and local associations. Your club will probably be entering something. Competitions definitely exist.
For the NSRA ones, you need to be a member of the NSRA to enter, which also gets you insurance (both liability and for your shooting kit), reduced entries to meetings (e.g. British Air Championships), subscription to the Rifleman, etc.
Entries for the winter season postal are due in August though, so get a move on.