When businesses or establishments put up signs that say, "No soliciting", they generally do not want people to come into that establishments for the sole purpose of selling something to the people inside. This is true for all the logical reasons. The product can be shady or the buyers may think the establishment/owner is the one behind the promotion or encouraging the sale.
So, technically, a bowler selling a ball, new or used, to one of his fellow bowlers (NOT someone who came into the bowling center SOLELY to sell a ball) is not soliciting. That said, the pro shop in centers are either owned by the center or charge rent to the pro shop. In either case, selling balls is competing with the pro shop (although many pro shops don't sell used balls, some do.) and the pro shop can ask the owner to throw you out or ask you not to return. It's within their legal rights for any owner of any establishment to refuse to serve or to provide services to anyone they wish, within normal legal boundaries, of course.