How can I ensure my AppRunner service can successfully access my DocumentDB?

Hello.

First, you need to connect AppRunner to the VPC where your DocumentDB cluster is located.
I think you can use it by configuring the DocumentDB cluster to be accessible from the security group and subnet you selected when setting up the VPC connector.
https://aws.amazon.com/jp/blogs/aws/new-for-app-runner-vpc-support/