Your best option here for using Raygun4Net in a web application would be to listen to when Raygun4Net is about to send a message, and then add any additional information you want to the message object before it gets serialized and sent.
Here is the documentation with a code example of how to do something like this: https://raygun.com/docs/languages/net/asp#modify-cancel-message
The code example includes how to provide your own RaygunClient instance to the http module if that's how you've set up Raygun4Net. (Let me know if you're not using the http module, or if you're working with WebAPI instead). This is done by changing your http application to implement the IRaygunApplication interface and return a RaygunClient instance from the GenerateRaygunClient method. When your app starts up, create the RaygunClient instance, and attach an event handler to the SendingMessage event.
Your event handler will be called any time an error report is going to be sent to Raygun, be it manually or automatically. In this event handler, you could either append to the e.Message.Details.Error.Message value, or you could set a list of tags to e.Message.Details.Tags, or even set a Dictionary of custom data to e.Message.Details.UserCustomData. (Ideally with null checks on each bit of those property chains). Just choose whichever works best for you, depending on where you want to see which app it was from.
Please let me know if you have further questions about any of this and I'll help you out.