So there's two things that may result this issue:

1 - you need to ensure that the IAM role associated with the API Gateway has the correct permissions to write metrics and logs to CloudWatch:

cloudwatch:PutMetricData
logs:CreateLogStream
logs:PutLogEvents

These permissions should be part of the policy associated with the API Gateway execution role.

2 - it may need some time for the logs to appear, did you tried to send many requests and then checked the logs?