Very good/focused suppliers can send you shipping info the same day you pay your invoice and send out your shipment the same day you send the supplier your labels.
On average it takes suppliers about 1-10 days to send you shipping info and then 1-10 days to ship out the actual product after the supplier gets your shipping info/labels.
This is all contingent on if the supplier has your product in their warehouse. If the supplier does not have the product in stock, it could take much longer and that has more to do with lead time than them packing anything.
If it's taking longer than 5 days from when you paid to get shipping info, and the supplier didn't tell you it will take a while, best practice is to follow up with your supplier every few days.