Depend on the agreement between the Overlord and the vassal.

An overlord can agree to join the vassal in all war, in offensive war only, in defensive war only, or none at all.

Likewise, an overlord can demand the same from the vassal, to join in all war the Overlord participate in, or only in either offensive or defensive war, or none at all.

You can see on the war demand page if declaring war on an empire would drag someone else into the war on their side or not.

Depends on the kind of vassalage they have. If it's a regular vassal then yes