API Gateway is a store-and-forward service. It collects the response from whatever the back-end may happen to be (Lambda, an HTTP server, etc.) and then returns it en block to the browser -- it doesn't stream the response, so it would not be suited for use as an Eventsource.
AWS doesn't currently have a managed service offering that is obviously suited to this use case... you'd need a server (or more than one) on EC2, consuming the data stream and relaying it back to the connected browsers.
Assuming that running EC2 servers is an acceptable option, you then need HTTPS and load balancing. Application Load Balancer supports web sockets, so it also might also support an eventsource. A Classic ELB in TCP (not HTTP) mode should support an eventsource without a problem, though it might not correctly signal to the back-end when the browser connection is lost. Both of those balancers can also offload HTTPS for you. Network Load Balancer would definitely work for balancing an eventsource, but your instances would need to provide the HTTPS, since NLB doesn't offload it for you.
A somewhat unorthodox alternative might actually be AWS IoT, which has built-in websocket support... Not the same as eventsource, of course, but a streaming connection nonetheless... in such an environment, I suppose each browser user could be an addressable "thing."