A quick google search seems to suggest that this is not possible at the moment
https://forums.aws.amazon.com/thread.jspa?threadID=205761
Re: Possible to use API Gateway for Websocket-based connections?
The lambda function has to stay active as long as the client is connected, which you don't want as you pay per 100 ms.
A quick google search seems to suggest that this is not possible at the moment
https://forums.aws.amazon.com/thread.jspa?threadID=205761