If I were going to create a virtual keyboard in Moodle, I might begin with the charmap plugin in the Atto editor, change the unicode for the character table to Cyrillic values, change geometry and CSS to make it look like a keyboard, and change the behaviour to allow clicking multiple characters. This would allow students to enter Cyrillic text in any textarea i.e. forum posts and essay questions using unicode. The only problem with doing this is that students would have to install unicode fonts on their devices for it to be useful.
A second idea would be to use Cyrillic support in LaTeX to create a keyboard in the Atto equation editor. This is easier since the TeX commands in the equation editor are set in the admin settings for the plugin. MathJax does not support Cyrillic yet so you would have to configure and use the TeX notation filter with Cyrillic macros loaded in the preamble. Students could construct words with the equation editor. They appear as TeX codes in the editor, but will be rendered as graphics in pages or posts. It will display correctly on any device.
For a quiz I would not be as concerned about students entering text via keyboard. You should be able to use multiple choice, matching of drag and drop for vocabulary and grammar exercises. Again you will need to consider whether Cyrillic unicode is adequate or whether you should use images or the TeX notation filter.