Where I live, you can easily get away with owning a horse and not having a truck. Local hay farmers around here deliver. Our vet comes to the farm for appointments, and our vet offers transport services, if our horse needs to be brought into the clinic for any reason. I have never had a problem hitching rides to shows, and finding transport to move my horse between stables.
Since some others say there is no hay delivery in their local areas, then it may be a good idea to figure out where you might be able to buy your hay and ask about transport. If you can find a few good, local sources that deliver, and if your vet has a trailer you can rent if your horse needs transport to the clinic, then you should be okay. Don't forget about bedding delivery as well.
Having a truck of your own would certainly make your life easier, especially if you plan to keep your horse at home, but it can be expensive to keep and maintain another vehicle, and if you have access to the resources you need to survive without one, then you don't have to have one, if it's not worth it to you.