Is Cypress ok for acceptance tests on APIs in a CI/CD pipeline?
Cypress is designed for front-end end-to-end testing.  It is designed for testing a users interaction with your application and verifying that it works correctly (e.g. logging in with a valid username and password load the landing page).  Tools like this are designed to allow running in a headless mode which will let you run the tests in CI tool without loading a GUI.

Since it appears that your goal is to test the API interaction you are probably better off using a tool designed for that purpose.  Most API test tools allow you to send a request to your API and verify certain things about the response (i.e., response code, response headers, response bodies).

While using an end-to-end tool (like Cypress) will ultimately test your API, it will take longer to setup, run, and often be more brittle.  To use your above example, a tool like Cypress will require a script/test steps that log into your system, selecting multiple addresses for shipping, and verifying that everything displays properly.  A tool like Postman would just test that when you send a request to your FedEx API with a specific body and headers, you get the expected response, body, and headers back.  Setting up something like Postman will require a new tool, but will ultimately save time 

Many of the API tools also provide a command line test runner and integrate with various CI tools (non-exhaustive list):


Postman - Free, relatively easy to setup, and includes an automated testing tool (newman).
SmartBear - Commercial product that integrates with most CI tooling.
Rest-Assured - Open source option for REST API testing.