Our site is small and has no rep. The council run it (after a fashion) and there is a waiting list, but I also know that the council has people at the top of the list for our area that turn down plot after plot, because it is not exactly what they want and plots they find very hard to let. Make sure they understand you are ready to tackle anything offered.
If there are sites where people gang together to clear overgrown plots for re-letting, you could offer to help. Likewise if there are community plots, a helping hand now and again might flag you up as a good prospect.
Just tread carefully. Our site is very laid back, but you do near stories of site politics and all sorts of upsets. Be polite, friendly and keen, but not pushy and see what happens