It's pretty simple, you just append your key to the URL as a parameter.
For example, looking for users 1 and 5:
So, it looks like your HTTP.call would become:
After re-investigating my code, I realized what was going on here. For this particular project I wasn't using the API key and access token of my own application; I'm using the ones I sniffed from the communication between the Stack Exchange mobile app and the API. The Stack Exchange mobile app has an unlimited quota, it seems.
See How API Keys Work. It's six years old, but a leading API developer said:
... If you can demonstrate a need for a higher request quota, contact us.
However, the Key limit should be sufficient for development, so please only request an increased quota when your application is live and has a non-trivial number of users.
(Emphasis and contact link ...
That error means that you have used up your API quota for the day.
You are allowed either 300 or 10,000 API requests per day for each IP address, or app, depending on whether you use an application key and/or an access_token.
The "Throttles" documentation
What are the API request limits? (FAQ)
Request Throttling Limits (FAQ)
Note that you must ...
See the Throttles doc page.
The 10K limit is not if the customer registers, it is if you register your app and provide the key when you make API calls.
Also, you can get more than 10K per IP if your app authenticates the user and then passes a valid access_token. In that case, it becomes 10K per logged in user and does not share the 10K IP limit.