Hank here from Shopify.
Currently, it is not possible to make any major changes to the online checkout for the basic plan. The main reasoning for this is due to a mixture of your protection, and checkout stability.
As the checkout is crucial to the success of your store, we want to ensure that no changes made to this can have a negative impact on your store.
That said, you can add Google Analytics to your checkout within Online Store > Preferences to help you with your tracking, etc.
You can also make some visible changes to some settings within your checkout within Settings > Checkout. Additionally depending on the theme you are using, you may be able to customize the look and feel of your checkout in your admin Online Store > Themes > Customize theme > Checkout, but outside of these, further customizations aren't generally possible.
While new content can't be added, you do have the ability to edit the wording that's already there. In your admin, go to Online Store > Themes, click the Actions button, and select Edit languages. In the next screen you'll see some tabs across the top of your screen; what's there exactly will vary depending on the theme that you're using, but you'd be looking for something indicating the checkout. This is typically a very long file, so searching it for what you need is probably a better approach than scrolling through the entire list. I'd suggest searching using command or control + F for 'checkout', and then looking for a field that contains that "OR" one that you'd like to change.
Let me know if you have any additional questions.
All the best,
The editing of the checkout area would be solely for the Shopify Plus plan as it is a part of the makeup of the plan as these stores operate differently from our core plans.
Then, depending on what API you are using, may need to be approved prior to use but as you mentioned you will be going through the API approval process. Or if wanting to use an app and wish for that to be approved, there is more information here relating to apps.