The most secure way to access an Amazon DynamoDB table from an application running on Amazon EC2 instances in private subnets is to use a VPC endpoint for DynamoDB. This ensures that the traffic remains within the AWS network and does not traverse the internet. Options B and C involve NAT gateways or instances, which route traffic through the internet, leading to potential security risks. Option D uses an internet gateway, which also exposes the traffic to the internet. Hence, the correct answer is A: Use a VPC endpoint for DynamoDB.