Tushar Bajaj, 35, and his wife, Leena, 42, have casually bought and sold stocks for years. Until last month, they had never been able to buy shares in an initial public offering. When Mr. and Mrs. Bajaj saw that Square was allowing merchants like them to buy shares in its November I.P.O. they bought about $2,500 worth of stock at $9 a share. Some 28 days later, those shares are worth about $3,500, up 40 percent. The Bajaj family was among 14,000 Square sellers that bought into the company’s I.P.O. through a platform known as Loyal3, the company is expected...