I'm using Amazon cloud drive to save many video recordings. I want to run my own program in the cloud to process and/or edit these files.
Is there a way to access the data in my cloud drive from a program running on an EC2 instance?
Yes and no. You can use Amazon Cloud Drive REST API to access files. To authenticate on Cloud Drive through your personal device you also need small web server to get authentication tokens. Main problem you need to get security tokens by registering your security profile and "whitelist" it. Since September "whitelist" became unavailable without special review which takes about a month and almost 100% you get rejected. Even documentation for Cloud Drive is not accessible on dev console page anymore.
I would recommend to reconsider using Amazon Cloud Drive.