For example, are these sentences correct?
The boys felt their chairs shake.
The desks and chairs shook.
(I mean it is nowhere mentioned who is actually responsible for shaking them.)
No, you can't make a sentence without a subject, unless it is a command (e.g. "Shake your body!"). In your second sentence, "the desks and chairs" is the subject. In this case, the verb has no object. Those kind of verbs are called intransitive.
If you look up 'to shake' in a dictionary, you will see that it can be used both as an intransitive verb (without object) and as a transitive verb (with object).
You missed a definition of "shake" that functions differently.
"shake" can be used as an intransitive verb meaning "for something to move or sway with short, quick, irregular vibratory movements" (Dictionary.com) or something like that. Intransitive verbs need subjects, but do not take objects.
The subject of the shaking in your case would be the chairs and desks. A verb cannot be without a subject, unless if it is imperative (e.g. "Shake it!").