Are players free to make someone from the order of the guardians? And what would that RP even be like I wonder. Also, I was trying to hint at that to take the actual monk class you don't need to be part of an actual order or monastery, more just adopt the practices, right? I.e. a person who prays on Ezra's teaching, meditates on his role in them, and contemplates how to live their life out, then lets say exercises and workouts out to sort of dedicate a healthy body to said deity, is essentially a monk in terms of the class, although there's no canon Ezra monk order. The RP makes sense and sort of opens up the limited prospect of playing the class, in my opinion at least
A "monk" in the real-world sense (both Western and Eastern Christianity, as well as Asian) is a religious ascetic, who lives a life of varying degrees of simplicity and deprivation. Some monks follow specific religious orders, but some do not (Eastern Orthodox is like this - they are religious and adhere to their church, but are not part of a formal order and do not necessarily follow formal rules, like the Rule of St. Benedict that some monks in Western Christianity do). Many monks are cenobitic (i.e. live in communities, like monasteries), and some of these communities may be cloistered (they do not go out, and do not let people in), but there are also monks that live as "anchorites" (i.e. hermits of a sort, they live in isolation -- sort of like solitary confinement). It is interesting that Ravenloft uses the term anchorite for something completely different.
In western religion, monks were generally stationary (affixed to a particular monestary or hermitage). Friars were the traveling version of a monk in Christianity. Asian monks are also generally attached to a specific monastery, and the idea of the "wandering monk" is more romantic fiction than reality -- Buddhist monks, for example, will wake up and walk around their village or city collecting alms in the form of food (which they are not actually allowed to ask for, technically, but people are expected to give them food). But the monks return to their monestary at the end of their day. The one exception are Jain ascetics -- their goal is to own no possessions (which is why they don't have monasteries) and they wander from place to place. In Buddhist teachings, religion is not only an individual endeavor but also a collective one, so it is rare that you would find a "monk" not attached to a monastery.
The D&D character class of "monk" is almost wholly derived from the stereotype of the "Shaolin Monastery" . It is an actual place, by the way. It is a Chan Buddhist monastery ("Zen" Buddhist, in Japan).
So to answer your question, yes you could in theory have a monk that is solo, but if you wanted to be historically accurate, they would be a hermit or something similar (of course, there is no rule that says you have to be historically accurate, although others might get picky about it). You could also have a group of monks that are not formed into a formal order (like Eastern Orthodox) or are unattached to any physical monastery (like Jains), but all monks follow either a religion or set of rules of one sort or another.