As Far as I know, there is no current implementation in flutter for Kinesis Video "Producer" SDK.
However there's Android Implementation, so what I would suggest is to add this Android Native Code in your project, and call it from Flutter Side.
The flutter camera library can be modified to work with Kinesis Producer SDK.
Or like @Andrija said, REST API as a proxy for Kinesis can be used. But the downside is audio won't be streamed, you might need to container (MKV/MP4) the audio and video and send that.
Having said all that, if you can somehow encode the video and audio (MKV/MP4) from flutter, then you can use putRecord
in aws_kinesis_api
flutter api to send it to Kinesis. But nowhere does it say it has to be video/audio, this is a generic api to put data into the stream.
EDIT: This confirms that as of current SDK, there is no Kinesis Producer code for Flutter https://github.com/agilord/aws_client/issues/242#issuecomment-860731956