Horses are expensive to own. Like any other pet, you must be able to provide for food, shelter, and care- however, since they are considered 'Farm Animals' you will have to have a vet who is licensed to work with them, you may also have to find a farrier to shoe the horse as well.
In addition, visiting the horse only on Sundays isn't truly taking care of the animal. For the other six days of the week your horse will be in the hands of someone else while you are only visiting or paying for that care. It isn't quite fair to you, the horse, and the person caring for it-- and with all due respect, it isn't exactly the same as actually *owning* that animal. Basically it'd be like owning a pet that will end up bonding with someone else, rather than with you.
Before you jump at the chance at wanting to own a horse, you should TRULY think about the reasons you want one. Just because they're neat to look at, are pretty, and you'd like to ride one every once in a while? Ask yourself if you are willing to muck out [clean] your horse's stall, feed it, groom it, exercise it-- OFTEN to keep it healthy. This will become your responsibility, not whoever is boarding the horse for you- and this will very likely take up a lot of your time if you only get to see your horse once a week.
If you're certain that this isn't just a phase you're going to grow out of in a few years, or because it will become too much of a problem to continue taking care of it-- then there are PLENTY of websites you can look on for horses in your area, or as close to as possible.
DreamHorse.com - Horses for Sale - Dream Horse Classifieds is a very notable site.
Petfinder.com: Adopt a pet and help an animal shelter rescue a puppy or kitten. is another, if you look under 'farm animals' or 'horses' or sometimes 'other'.
Please do think this through before making a hasty decision.