As far as I'm aware, it seems to be the norm for British wizards to have their own Gringotts accounts. This goes for the great and the good of society, the rich and the poor. (If you think that people who are believed to be Dark Wizards can't use Gringotts, think again). Yet Voldemort didn't have money as a child. It seems that the penniless Merope didn't leave him anything since Hogwarts had to give him a grant to attend.
When he had finished he turned to Dumbledore and said baldy, "I haven't got any money."
"That is easily remedied," said Dumbledore, drawing a leather money-pouch from his pocket. "There is a fund at Hogwarts for those who require assistance to buy books and robes."
(Half-Blood Prince, Chapter 13, The Secret Riddle)
But surely when he got older Riddle would have got an account in which to deposit his wages from Borgin & Burke, and any savings. He may not have cared much about money but a bank account is a fairly basic part of being a grown-up. Yet Harry isn't at all confident that he ever went to Gringotts.
"I don't know whether he was ever inside Gringotts," said Harry. "He never had gold there when he was younger, because nobody left him anything. He would've seen the bank from the outside, though, the first time he went to Diagon Alley...I think he would've envied anyone who had a key to a Gringotts vault. I think he'd have seen it as a real symbol of belonging to the wizarding world."
(Deathly Hallows, Chapter 24, The Wandmaker)
It may make sense for the child Riddle to envy people who have Gringotts vaults. But why would the adult Riddle still think that way, years later, when he came to think of hiding places for his Horcruxes? Do we have any reason to think that Gringotts doesn't allow people in the wizarding community to open new accounts when they don't have a pre-existing family vault? (Adamant argues - sensibly, I think - that we don't here).
Furthermore, if he did have a vault of his own, why use the Lestrange's vault to hide the cup? Did he perhaps consider the Lestrange vault more historic and prestigous and therefore worthy of a Horcrux? Or is the fact that he asked Bellatrix at all an indication that he didn't have a vault of his own?