Seriously, any vehicle that can take 13 to 14 passengers has to be a public service vehicle , a bus or coach.
Health and safety regulations will not allow a farm cart or trailer to be used for passengers. No fire engine or such like can legally carry 14 passengers either, No chance of using a commercial truck and all piling in the back of an Eddie Stowbart, again not licenced for passengers , health and safety will be on you like a ton of bricks.
Good ideas for say 3 or 4 people but you are seriously limited when you look for transport for 14.
Is it near a railway station? hire a steam train !
i have had 2 of my kids do the 'Prom' my daugher shared a limo and my son walked there.