Can The Apple Store recover data from an iPad Pro that turns on and off because of water damage?

“Will the Apple Store be able to save the data from the local app?”

no.

Apple doesn’t do data recovery at all and they will not touch a water damaged device.

Your data is your responsibility.

You should have been making regular online backups of your data if it was really important.

If you have a pre-existing iCloud or iTunes backup of your iPad, that was created prior to your iPad becoming unserviceable, the backup (that includes locally stored data) can be restored to a replacement device:

Restore your iPhone, iPad, or iPod touch from a backup - Apple Support

If you don't have a backup, your data may irretrievably lost. Your iPad has an encrypted filesystem. Unless the iPad can be restarted to a running state - and you successfully enter your device Passcode - all data remains encrypted and beyond reach.

Successful entry of the device Passcode unlocks the Secure Enclave (the iPad's security chip) within which the only copy of the encryption keys, required to access locally stored data, is stored. When the iPad is shut down, or restarts, the encryption keys are purged from RAM until the iPad is next successfully unlocked.