I sort of have this issue. I don't have for rent but my sim lives in the Feng apartment in San Myshuno. She is an artist and when the painting needed to be shipped it takes several tries to get her to do it. But eventually she will. The best way for me is have her stand right in front of the mailbox before I ask her to ship it.
I'm having this issue in an apartment in San Myshuno. No mods or CC. All packs installed except a couple of kits, and playing on Windows 10. Game version 1.106.148.1030.
My sim has sold some knitted items and they are marked in blue ready to ship to the customer but, no matter how many times I click on 'Ship on Plopsy' the sim doesn't do anything. Nothing comes up in her queue. Nothing happens at all.
I tried putting testing cheats on and resetting the apartment mailboxes but it didn't help. Just in case it makes a difference, I am playing in 20 Culpepper House in the Spice Market.
First time with a sim selling on Plopsy here!
She lives in an apartment in the San Myshuno Spice District, so can't replace the mailbox.
My game is behaving like those of @reboede and @Gabbywolf where the first couple of attempts drop from the queue, and eventually one works. I click "Ship to buyer" anywhere from 2 to 10 times before it works. I don't think I'm doing anything differently between attempts.
