Do you plan to move backend from Wordpress to Frappe?
If you can change code of this app use OAuth 2.0 for making authorized requests as the user instead of removing Authentication.
https://frappe.io/docs/user/en/guides/integration/using_oauth
if your app is hosted on another domain you may encounter issues related to CORS
refer Access-Control-Allow-Origin Error for jquery $.ajax call - #11 by revant_one