You can request a specific sponsor. My cadet and the officer to whose house she went for the Ice Cream Social "hit if off" and they requested one another for sponsorship. I think that is the caveat; both parties must request in that instance. I do know of a few friends of my cadet who had prior ties to officers on post and requested one another. I know of one cadet whose sister had a tie to an officer, so that officer requested the sibling as a cadet.
You will also find that over the course of time, you and a family may unofficially "adopt" one another. For example, my cadet's sponsor also includes several of her friends, whom they truly enjoy, when they hold group get togethers. There have been several officers on post with whom my cadet has become close and although two of them have moved on, she still keeps in touch with them regularly.
One thing to keep in mind with your sponsor is that it is a two way street. If your sponsor says to call if you need anything, take them at their word and call them. They can be lifesavers, giving you rides places, letting you do your laundry, having you over for Sunday dinner, etc. My cadet's sponsor family has taken her to Broadway several times and has actually let us use their house for weekend visits when they happened to be out of town. Above and beyond what is expected, but I have heard that is not an uncommon practice.
On A-Day, her family invited us to their house to wait until she could join us, rather than us waiting under a tree or by a statue. They also took us on a tour of West Point unlike one you would get with a tour company.
So use your manners, say please and thank you, talk to them and be conversant, and you can form a lifelong relationship with some great folks! This family has been an integral part of my cadet's life for 3-1/2 years; it will be hard to tell them good-bye in May, although I know we will stay in touch with them, as will she.