Is that really true? Section 51 of the London Hackney Carriage Act 1831 (to which the article refers) doesn't seem to require that; it (among other things) forbids a taxi driver feeding another driver's horses with anything other than corn or hay - but doesn't say anything about requiring them to carry hay for this purpose. There is no other reference to "hay" in the act.
https://www.legislation.gov.uk/ukpga/1831/22/pdfs/ukpga_1831...