your goal is not clear.
the relay/hybrid is typical proxy application with two components. the azure server proxy, and the on-premise proxy client. the proxy client connects to the azure server so, that no inbound connection is required. the sql client code, just connects to the proxy, which behaves just like a direct connection to the sql sever
if you reversed engineered the proxy protocol the you could implement the server code in your application. this would require your site have an additional ipaddress for the proxy connection to on-prem, so you'd have to use a VM to host your site. this additional ipaddress is what sql client code would connect to.
instead of reverse engineer, it would probably be easier to just write you own proxy server and client. As sql client does not support http, the proxy code would need to be hosted by a VM to support raw tcp traffic.