The answer by Rafael focuses more on using the Websocket API Gateway which in my opinion is still relatively new and there is some space to improvements. Plus I don't like having lambda integrations with database access because without RDS proxy they exceed the db connections really fast, and I don't think HTTP integration adds anything to the whole thing because you're performing HTTP request in the end but it's called through the Websocket API.
One thing I agree on with Rafael is that you need to have an EC2 instance running socket.io whether it's in Node.js or python (I used python with Flask in my case).
I managed to connect to my socket.io by using the HTTP API Gateway and setting allow_upgrades=False
so http protocol won't be upgraded to ws protocol, because HTTP API Gateway doesn't support ws. My HTTP API Gateway is just forwarding socket.io requests to the load balancer, and good thing about that is that you can define access control on each route defined in the HTTP API Gateway.
The socket.io on my EC2 instance is defined like this:
socketio = SocketIO(async_handlers=True, allow_upgrades=False, cors_allowed_origins='*')
And my client connects to it by simply calling the route defined in the HTTP API Gateway which has proxy integration enabled.
https://xxxxxxxxx.execute-api.us-west-2.amazonaws.com/socket.io/{proxy}
Final result - client connected to socket
