I am writing this for use in the Midgard setting, but I should state upfront that this can be used for just about any setting. A grappling style Monk would fit in just about any setting that allows monks. I recently wrote a blog post about the weird misconceptions held about martial arts and how they shape their presentation in D&D. One of the reasons I wrote that was because I could see some specific archetypes for Monks in Midgard I wanted to make.
So, there is an idea out there that monks have no place in a western European inspired campaign setting. Settings like Forgotten Realms or Greyhawk owe a lot to western European history, legends, and literature for their basis, and monks are not part of those traditions, or so the argument goes. This came to mind while looking at the Midgard setting and noticing they had many character options for the range of character classes, but not for monks. I asked about this and was directed to this blog post about monk weapons characteristics, which is awesome, but was also told there was not a lot of places for monks in Midgard to be from. That feels like something I can counter. Note: I love the Midgard setting and I am not offering criticism here. I am writing this because I feel this is a common sentiment that maybe should be countered.