Well, there is a way to add a datasnapshot to your free cloud environment.
When you have created a data snapshot of your local database, then on the first deployment to a free cloud environment the data snapshot will be pushed to the cloud. This is equal to local deployment. When you doesn't have a local database yet, the data snapshot file will be used (if existing).
If you have already published your app to the free cloud then this mechanism doesn't work anymore.
Unless… you unlink/delete your free cloud and publish your app again. ;-)
When publishing again, automatically a new free cloud environment will assigned.
So short answer on your question; Yes it is possible
Data snapshot is only used for built in databases. You can only use these among developers locally. Mendix uses postgresql on cloud.
You may require some sort of import mechanism to refill your data on free cloud as free cloud does not allow to upload databases manually (even if you had postgresql setup on your local machine which you could’ve exported).
If there is not much data, doing it manually might also be an option.
There is no shortcut here. ‘Some import mechanism’ that Radhika suggests are your only option. They require you to create download- and upload-mechanism. Or another option: create a JSON-string that you include in the code and run through an import-mapping after startup. Things like that.