Hurdy Gurdy by Christopher Wilson describes a very small one-monastery order as the setting for the story at the time of the Black Death, existing alongside the much bigger Benedictine and Cistercian orders. Was it common for there to be very small scale monastic orders or did a lot of them get absorbed quickly into bigger orders?
In short answer, it was theoretically possible [to form an independent order] in the single monastery especially until the first half of the 11th century, though it might not be so realistic.
The main problem to offer the simple and definitive answer to OP's question is the ambiguity of the use of [the monastic] order (ordo) both in contemporary (medieval) texts and in modern literature, regardless of academic/ popular division. On what criterion the alleged medieval order could be regarded as distinct?
The majority of "Benedictine order" almost throughout the Middle Ages didn't in fact necessarily have an institutional basis beyond individual congregations - what they essentially had in common was the Rule of St. Benedict. Famous Cluniac monastic networks, led by the monastery of Cluny (founded in 909/10) and extended mainly across SW Europe, was certainly well-organized (in a confederation), but regarded primarily as a sub-"order" within the Benedictines since they didn't compile another "Rules" for themselves.
It was not until 1893 that the whole Benedictines across different confederations like Cluniac one finally had a unified leader of the order under the instruction given by the papal brief Summum Semper.
So, was a distinct monastic rule (not just a slightly modified version of the Rule of St. Benedict) enough to form the independent monastic order?
Then, the congregation would probably require the superior church authority that guaranteed the foundation of monastic order with a distinct rule in form of the special document (foundation charter) against the possible criticism within and out of the local church, especially in Later Middle Ages. Such an authority could be the local archbishop (I suppose it was the patriarch of Jerusalem in the case of the Carmelites), but preferably by the Pope himself.
It is well-known that the Society of Jesus began just with 7 members (including the founder, Ignatius of Loyola himself plus 6 other members) in 1530s and succeeded in getting the papal approval in a few years later, 1540, but I'm not sure whether OP's congregation could enjoy a good connection with the Pope or at least with the local archbishop (who might in turn recommend the request to the Pope, with eager support), as were the cases of Ignatius or Francis of Assisi (in other words, it is likely that such a new congregation with good reputation could be more popular in form of more than a few members).
In case of the first millennium (without the strong church hierarchy in reality), such a document would not have been obligatory, but the concept of a distinct order besides the Benedictines itself had not been so common.
Reference: