You are correct in needing a "Forward Proxy". A good analogy for this is the proxy settings your browser has for outbound requests. In your case, the web application behaves like a desktop browser and can be configured to make the resource request through a proxy.
Often you can control this for individual requests at the application layer. An example of doing so with C#: C# Connecting Through Proxy
As far as the actual proxy server: No, it does not need to run Windows or IIS. Yes, you can use a proxy service. The vast majority of proxy services are targeted towards consumers and are used for personal privacy or to get around network restrictions. As such, I have no direct recommendations.
Cloudflare actually has recommendations regarding this: https://blog.cloudflare.com/ddos-prevention-protecting-the-origin/.
Features like "upload from URL" that allow the user to upload a photo from a given URL should be configured so that the server doing the download is not the website origin server.
This may be a more comfortable risk mitigator, as it wouldn't depend on a third party proxy service. A request for upload could be handled as a web service call to a dedicated "file downloader" server. Keep in mind that if you have a queued process for another server to do the work, and that server is hosted in the same infrastructure, both might be impacted by a DDoS, depending on the type of DDoS.
Your question implies that you may be comfortable using a non-windows server. Many softwares exist that can operate as a proxy(most web servers), but suffer from the same problem as ARR - lack of support for the HTTP "CONNECT" verb, which is used by modern browsers to start an HTTPS connection before issuing a "GET". SQUID is very popular, open source, and supports everything to connect to.. anything. It's not trivial to set up. Apache also has support for this in "mod_proxy_connect", but I have no experience in that and the online documentation isn't very robust. It's Apache, though, so it may be worth the extra investigation.
