It looks like a duplicate of old issue https://github.com/localstack/localstack/issues/78
But now it's reproducible for me with the latest version of LocalStack. 
resources = {}
        template = self.APIGATEWAY_DATA_INBOUND_TEMPLATE % (kinesis_stream)
        resource_path = self.API_PATH_DATA_INBOUND.replace('/', '')
        resources[resource_path] = [{
            'httpMethod': 'POST',
            'authorizationType': 'NONE',
            'integrations': [{
                'type': 'AWS',
                'uri': 'arn:aws:apigateway:%s:kinesis:action/PutRecords' % DEFAULT_REGION,
                'requestTemplates': {
                    'application/json': template
                }
            }]
        }, {
            'httpMethod': 'GET',
            'authorizationType': 'NONE',
            'integrations': [{
                'type': 'AWS',
                'uri': 'arn:aws:apigateway:%s:kinesis:action/ListStreams' % DEFAULT_REGION,
                'requestTemplates': {
                    'application/json': '{}'
                }
            }]
        }]
        return aws_stack.create_api_gateway(
            name=gateway_name,
            resources=resources,
            stage_name=self.TEST_STAGE_NAME
        )
    def connect_api_gateway_to_http(self, gateway_name, target_url, methods=[], path=None):