Connect and plumber certainly look like interesting options, but you say it is possible using our existing Shiny infrastructure? I haven't been able to find any documentation on this.
To say for sure, I'd need to know more about your project, but you can use shiny to generate and serve assets that other programs can request (i.e., act like a web service). The plumber package is designed and optimized for that sort of task. I highly recommend watching this talk.
You might also consider just varying the logic in your shiny app based on the URL parameters given to the application.
I think a critical question to consider is, who is managing the input values (i.e., the input parameters that generate your graph), you or the web developer? If you need to manage them, then you likely want to use the resources I point to above. If your web developer is managing them, you're probably better off hosting (plumber-based) HTTP endpoints that the developer can request.