An API key is tied to a domain. So, you have an API Key for any custom short domains you have created, and one for budurl.me.
To get the API Key for your custom domain, login to your account and make sure to select your custom domain in the header. Then go to Settings at the top of the page, select “My Account”, which is under the “Personal Settings” header. This takes you to a profile page, and you’ll find your unique API key at the bottom (the actual key is blurred in the screenshot for security purposes):
Next you’ll need to integrate the API on your end. You can find details on API documentation at this link to pass on to your Development Team. Let us know if you need any assistance!