Can I install Mac OS X on a blank SSD without any backups?
If you have a bootable USB stick yes you will be able to install it onto the new SSD by holding down the option key at startup and selecting it.

The command to create an installer USB varies slightly on the version:


https://support.apple.com/en-gb/HT201372


Alternatively your Mac mini is recent enough that you use macOS Recovery over the Internet