I probably share the frustration of the person you refer to as vexatious.... As I said, I am familiar with this scenario.
The answer to your question is NO, there is nothing you can, or should, do in relation to a Cllr asking questions. Well, you 'could' try and have a long hard think about what it is that is being asked and why it is seemingly such an issue to have someone asking questions....
Post up 1 example of a vexatious question and the answer which was provided to it. I'll wager it will be the answer rather than the question where the problem lies....