Can I remove Catalina and install Mojave without Time Machine?

You can, but... why?

The Mojave installer can be downloaded from Apple (you can search for it online, it is not easily found in the App Store).

You can do a Time Machine backup of your Catalina system so you do not lose all your documents.

Create a bootable Mojave installer on a thumb drive, erase and install and then migrate your user account from the backup.