I'm not familiar with the term sponsoring in the context you are using. Do you mean that you would fully lease a horse and then someone else would pay to ride it once or twice a week? If so, I know that depends on your lease contract with the owners. Some owners do not care, and some are very specific about who is contractually allowed to ride the horse. I hope that helps.
I did this with a horse I leased once. It was something I specifically asked the owner about before I started the lease and was written into our contract.
I wouldn't try it if you would need the sublease to help pay costs; there's no guarantee you'll find someone to sublease and no guarantee on how long they'll stick around if you do.
If you end up buying your own horse you can, of course, do a partial lease to someone else, but consider if you're really going to want some stranger riding your horse. Now that I have my own horse, I let my trainer know that if the right person is looking to partial lease I'd consider it (he could certainly use more exercise!), but I'm not really looking hard for someone to sublease and would be pretty picky about who I allowed to do so.
It has to be agreed upon with the owner.
I part lease a horse, and with the owner's permission, I have occasionally allowed someone to ride him and asked for a small sum of money to help me pay my part lease.
I agree that I would not want just anyone to ride a horse if I owned it, but having someone you can trust ride your horse and keep an eye on it can also be a reduction of stress, because you know that there are 4 eyes on it now, and it is getting regular care and excersize, even if you cannot get out there as often as you'd like.