"Because we need to burn the Chametz the next day we use wood and paper, which are flammable materials."
The website itself does not cite any further source.
This article on Aish.com clearly says that one is not required to use a candle - a flashlight is acceptable. Since it makes no mention of a spoon at all, one may infer that a spoon is not required either - and certainly not a wooden spoon.
If one can indeed fulfill Bedikas Chametz without a wooden spoon, our questioner is asking, "Why use a wooden spoon?"
That, I can answer in ONE WORD...
A fellow in Shul this morning told me that Rabbi Meir Shapiro ZT'L brings down in the name of the Rashash Mesechtas Shabbos 66 that since per Rabbi Yehuda, Nosar has to be burned with a fire fueled by wood, the Chometz must be burned the same way, with a fire fueled by wood. I asked him that answers the wood part however it still does not explain why it has to be a spoon? Why not a twig or a 2 x 4? He said he saw an answer and does not remember and will let me know. So I will add that part in later.
The Shulchan Aruch haRav writes (OC 445:7)
ואם לא מצא שום חמץ בבדיקתו טוב לשרוף הכלי שלקח לבדיקה כדי שלא ישכח חובת הביעור לשנה הבאה וגם כדי שלא ישכח לבטל ביטול השני שנתבאר בסי' תל"ד:
If one didn't find any leaven in his search, it is appropriate to burn the vessel which he search with [the next morning] in order that the process of burning [on Erev Pesach morning] not be forgotten in future years, and also so that one does not forget to nullify [one's leaven] the second time [in the morning].
Perhaps one could speculate that those who use wooden spoons do so in case they don't find any leaven and want to burn the spoon the next morning.