Ash here from the Shopify team. Welcome to the Shopify Community!
I'm happy to help out with your inquiry! I see that you're looking to integrate 3rd party shipping to your store, but could you please elaborate on exactly what you're looking to accomplish? Is your goal to connect your own third party carrier account so that you can use your own rates from them? Or perhaps you're looking to connect a third party shipping application?
Feel free to reply back here and let me know your end goal! I'll be happy to further assist once I have some more information.
Thank you for the additional information on your end goal. I did some digging for an existing integration between shipcompliant and Shopify, and it doesn't appear that there is anything available currently. With that said, I would try contacting a Shopify Expert to see if they can create this type of functionality for you. Shopify Experts are trusted, third-party agencies and freelancers, who are available for hire by Shopify Merchants. You can visit their site here to contact an Expert, or post a job request and receive quotes from eligible Experts. This would be the best direction to take at this time, as there are many Experts who specialize in creating custom solutions via API.
This is an accepted solution.
You can use the carrier service API to create your own carrier
You will need to create your own web service. Which users call and feed it certain info.
Your web service returns eligible shipping methods which in turn is displayed on the page.
When creating a custom carrier you specify a url to call your web service.
Take a look at creating a custom carrier. The bulk of the work will be creating your own web service that calculates your custom rates