Cosmoteer does not save ships to its installation folder, it saves ships to Documents\My Games\Cosmoteer
. If that doesn't exist at all, then there's something on your computer blocking Cosmoteer from saving files to your Documents folder. I've never heard of this being caused by anything other than an anti-virus or anti-ransomware software.
If you can't figure out what's blocking Cosmoteer, you can still tell Cosmoteer to save files in a different location using a command-line argument: --rootsavepath C:\Path\To\Cosmoteer\Saves
(replace the path with where you want it to save files)