Some sites do not allow holders to keep any livestock so you will need to check the small print of your contract.
My site has 20% of plots allocated for livestock, all are predesignated. This 20% is a combination of hens and pigeons. If you want to keep hens you can work another plot then swap plots later if you so wish. With waiting lists as they are people can be waiting up to 2 years for an allotment or in excess of 4 years if they want a plot to keep livestock.