Why is asking your users to log in not an option? If they are using a client-side app then it is reasonable to ask that user for permission to make API requests on their behalf (ie, get an OAuth token from the Implicit OAuth flow, which is design for client-side apps).
There are also free/low cost servers or serverless infrastructure available such as that offered by AWS that can handle an OAuth flow and deal with an API requests on an on-demand basis so you don’t pay anything except for when you actually make requests and even then they have free tiers on many of their services.
If you’re still not able or willing to do either of those options then yes it looks like you will be out of luck. Requiring accountability for who is making requests is not something that is out of the ordinary, most services online that offer 3rd party APIs usually require some for of OAuth token or similar authentication token to access their API endpoints.