I don't think they need to ship the cartridges themselves. Customers can just go to any store selling printer ink and get the cartridges. It's a bit steep to also need to buy the money-making product HP wants to earn on, but you can just refill them with cheap ink when they're empty so it's not as big an issue in the long term.

HP could stop selling ink cartridges to stores, but they'd be shooting themselves in the foot as well.