What service do you use for your shipping? Most have the ability to input the customer email and phone and when the order is complete it will email (possibly text) the customer the tracking information. Personally, I think it's better to utilize this service from the shipping provider.
We use Shippo, which is a partner with square. For your online orders, you can import the orders directly into Shippo and keep track of everything more smoothly.
Most of our shipped orders are taken in-person and do not import directly to Shippo. Currently, I just copy the tracking number and type in the creation date into a note under the customer info in square just as a backup and helps me know quickly if I shipped an order to that customer.
There are other services, but none that I know that work directly with Square (for online orders).
Shipstation for instance, used to partner with square, but wasn't listed last I looked. It was a paid service, very robust. Was honestly more than we needed.
Shippo is currently free, you just pay for your postage. Not sure if that is going to change, but at least until the end of the year.
The only other option I can think of is to utilize the customer messages and copy the tracking info with a note and either email or text it to the customer.
