Finding real organizations based on the hunting of monsters and other supernatural beings would be difficult as most of these "hunts" would be temporary events based on public hysteria due to a natural incident (mob-based witch hunts of history) or were more individual efforts like hanging garlic and paying attention to signs (like a 2012 vampire scare). Mythological is not going to be much better as the typical stories are based on "errant knights", the lone hero, or just small groups of adventures (Gilgamesh involves only 2 people & Journey to the West only a small group, for example).
That said, I do have two ideas that could be altered to fit (or even be considered) monster hunting organizations:
Of "demons" & Yokai:
Classical Western Demons
You already are using the Order of Inquisitors so why not include the Order of Exorcists? Exorcisim is not technically monster hunting in the strict sense but could also be adapted.
There are also two organizations which definently exist for this purpose. The first is The International Association of Exorcists (not affiliated with the Catholic Church1) which was founded in the 1990s with the focus of finding and stopping demon possession. Also, though there is no Order of the Exorcist within the church, the Vatican does require that "each diocese have a specially trained priest who is able to diagnose demonic possession and perform exorcisms when necessary."
Eastern Spirits and Demons
In Japanese culture, yokai, akuma, and oni tend to mix and merge so its even easier to see them as "monsters". In this case though, the Shinto religion places such emphasis on purification that any priest (Kannushi) of the sect can perform these.
In modern times, the idea of a priest purifying an object or area to rid it of yokai invokes more of an image of water, salt, and a old man waving a bunch of white paper. In ancient times, stories of people actually battling these creatures certainly exist and we have records of shinto priests being employed by the Empire to perform purification rituals as early as 772. So in that sense, these early temples could be seen as monster hunting - though more as a by-product of serving their kami than as a real focus - and if an area happened to have above average yokai/oni/akuma rate a temple might dedicate a portion of the priest and meka as purifiers.
1 I did find a reference to the removal of the actual order in 1973 but the source is suspect - so unless I find better evidence I wouldn't say the Catholic Church had an official order of exorcists.