I just replied to a similar & will repost here.
I had this problem for ages, Turns out I was trying to customise my character's plate in slot 2 (I didn't have one in slot 1)
I made a character in slot 1 and played up until it says you are on your own, Sent custom
order for plate and paint job.
No text still so I quit the game and restarted, When I got back online my order had been done, plate and car had changed (no need to visit LSC this time for me).
So I changed to my main character (slot 2) and sent an order for a custom plate and I got a text almost instantly, All my car mod orders have worked online since.
Check that you are ordering for a character in slot 1.