I'm not sure actually. They seem serious about it.
Well the idea of a coup isn't new. We have heard this before, mostly coming from far-right movements with close relations to the army.
During the demonstrations against gay marriage, many children from Army officers were arrested because those families were involved in the protest. An organization called "Le lys noir" called for a coup, and the Ministry of Defense requested an investigation. It was said that they had a list of names.
In this case, it's hard to tell without further investigation whether this movement is a "citizen" movement or far-right related. They claim to be apolitical, they talk about the Declaration of the Rights of Man, and about democracy. But there is nothing that the far-right couldn't have said.