yes, you can install OpenPetra on your own (Linux) server.
You must have root access though. Have a look at https://get.openpetra.org/
You must also maintain your own updates, etc, therefore it would be good if you have your own developer contributing to OpenPetra.
Please note: we have recently added the functionality for sponsored children, and the functionality for supporters to check their own data, but we have not yet a combined functionality, for supporters to check the data of their sponsored children. You are welcome to contribute this functionality.
Alternatively, I offer the service to host OpenPetra for you, and to do development for you, at reasonable rates.
But again, OpenPetra is open source software, and you are very welcome to have your own developer and system administrator, and they will get free help here in the forum. I really want to see a community that grows.
All the best,