Hello,
As you may know, Shopify only allows you to set up a default box and it doesn’t work for merchants having products with different sizes.
So, if you want to handle this directly in Shopify, you will need to manually select the box based on the order.
For automating the process, I would suggest you try the Multi-carrier shipping label app, which helps you set up different packages for your products based on the weight and (/or) dimensions.
So, whenever your customer places an order, the app automatically chooses the right box and calculates the rates accordingly. Also, with the app, you can print labels, and also automate the tracking process.