Some info here:www.geneanet.org
You have to register, but it's totally free. Then search the name Vervet and select the one situated in Aisne (the department encompassing the old diocese of Laon).
After which, for more info:
- you could try to identify Edme Vervet's regiment (the presence of an officer called Mr Celeron being a clue) and then search its history and registers, at the French Service historique de la défense: http://www.servicehistorique.sga.defense.gouv.fr/
- you could check the parish registers at the Archives départementales de l'Aisne, or via the local genealogical society, the Cercle généalogique de l'Aisne.
(but I don't know if any of them works by mail)