Technically HTTP represents a bidirectional stream of arbitrary data in both directions, which follows a set of headers and is optionally finished by a set of footers. This is for example quite obvious when you look at the HTTP/2 specification. There is no need for the bodies to be sent in a particular order (response body after request body) or in a particular encoding (form-data, chunked, SSE, etc.). It can be two arbitrary byte streams.

This functionality is exposed by lots of HTTP libraries, e.g. the Go, node.js or C# HTTP libraries will allow to simply read or write to a request/response body stream just like to can write to a socket. As a proxy - just copy the data from one stream to the other.

What are are probably thinking is that the browser environment does currently not allow to use HTTP for arbitrary streaming. Instead they defined some fixed use cases and encodings for these use cases. If you use SSE encoding you can't send bytes without overhead anymore and can only do streaming in one direction. But: You will get a nicer browser API for retrieving the data. Using HTTP bodies for arbitrary streaming will be allowed in future revisions of the fetch API (which e.g. give you a ReadableStream).

Other HTTP libraries also do not allow for streaming but expect the body in either direction to be a fixed length thing. This will allow to represent the body e.g. as a "string" or "byte[]" instead of a Stream from which the application has to read. Thereby the API gets simpler, but not all possible use cases are enabled.