Check if your Textbox is working fine.
You can check this by entering some text on a product with Textbox integration and adding it to cart.
If the Text is now overtaken into cart and afterwards shown in the checkout, your Textbox is working fine without errors.
Shopify will then automatic show the Text in the order section.
If this is not the case, something is wrong. See here possible errors.
Drop us a message for help.