I am planning to run the repository server on a headless Debian machine.
On reboot, I need to manually reconnect to repository and then restart the server. Is there a way to automate this process, so that the server comes back up on reboots.
It is not really kopia issue but rather your OS - effectively you are asking how to start program (kopia server in this case) after reboot.
There are two most obvious options:
old fashioned one - use
@reboot schedule - Crontab Reboot: Execute a Job Automatically at Boot | phoenixNAP
modern (and I would recommend this one) - use
systemd - Use systemd to Start a Linux Service at Boot | Linode Docs
I included two links but you can find plenty of examples and guides on Internet.
If you make it working it would be nice if you share your solution.