Hello,
This is a straight-forward issue that I can investigate on your system if you like. Can you give me admin access to your WordPress system?
If so, please email me with:
1. link to your site
2. admin login credentials
3. the order_id of one order that you see on the database but not in the API
If you do not want to give me access, then I can guide you to debug this.
Essentially what I would do, is use wp-console to first check if the PHP-API reports the order, and then the JSON-API. Tracing the execution path of the code that handles the JSON-API will make it clear whether stale transients are used, or what is going on in general.
Let me know how you want to proceed.
with regards