IAM role not clear for connecting API Gateway and DynamoDB

The trust relationship for this role is for replication.dynamodb.amazonaws.com and not for apigateway.amazonaws.com.

You should go to the IAM console, create a new role, choose API Gateway and later add the appropriate policy to write to DydnamoDB.

Eventually your role should include the following policies:

{
    "Version": "2012-10-17",
    "Statement": [
        {
            "Sid": "VisualEditor0",
            "Effect": "Allow",
            "Action": "dynamodb:PutItem",
            "Resource": "arn:aws:dynamodb:eu-west-1:xxxxxxxx:table/Comments"
        }
    ]
}

and

{
  "Version": "2012-10-17",
  "Statement": [
    {
      "Sid": "",
      "Effect": "Allow",
      "Principal": {
        "Service": "apigateway.amazonaws.com"
      },
      "Action": "sts:AssumeRole"
    }
  ]
}