Likewise, anybody is free to look in my shed, which is never locked, but I can't help wondering what your warden expects to find in the sheds, and what he then plans to do about it. I can't imagine a rat living in a shed, unless it is stacked full of clutter. If you use your shed regularly, no rat would be interested. They are very likely to visit though and will return if you are leaving out food for them. If chicken pellets are the attraction they can easily be stored in rat-proof containers, but is your warden really going to police how you store your chicken pellets?
The rats on our site mostly live in the piles of rubbish some people like to build at the end of their plots or on disused allotments. We cleared two skip fulls of rubbish from one unused plot and found five or six old rats nests. There is rubbish piled to chest height on the plot next to me, near where I keep my hens, and rats occasionally visit from there, but have never raided the pellets in the shed. Its just not in my interest to be feeding rats rather than hens, so the pellets are stored securely. The best feeding for rats on our site comes from bird feeders people put up for wild birds. The cheap food has lots of wheat in it which the small birds don't eat and they just throw it on the ground where the rats come along and feast. What exactly is the extent of your "huge rat problem"?