AMIs are immutable. Every time you start an instance of an AMI on EC2, the local disk is in the exact state it was when you created (or 'bundled') the AMI.
That said, you can attach persistent storage (EBS) to an instance, or ship things off to S3 (vai s3sync, etc) or attach to a database (RDS), in order to have data that lives past the instance lifetime.
But none of this is enabled by default.
I believe it would be possible to write a script that executed on boot and attached to an EBS volume, (or pulled a previously generated file from S3, or attached to an RDS instance). If you put that in the image before bundling, it would execute every time you started up. Here's a recipe I found for creating and attaching an EBS volume on boot (I didn't test it; caveat emptor).
That's probably as close as you can get to what VMWare provides with EC2.