|Not the white lady the order probably had in mind ...|
The order was what was called a "votive order". Votive orders were formed on a vow, and were really a less serious form of "fraternal order", which also involved a pledge. Wikipedia lists the following fraternal and votive orders:
Compagnie of the Black Swan, founded by 3 princes and 11 knights in Savoy (1350)
Corps et Ordre du Tiercelet (Corps and Order of Tiercelet), founded by the vicomte de Thouars and 17 barons in Poitou (1377–1385)
Ordre de la Pomme d'Or (Order of the Golden Apple), founded by 14 knights in Auvergne (1394)
Alliance et Compagnie du Levrier (Alliance ad Company of the Greyhound), founded by 44 knights in the Barrois (1416–1422), subsequently converted into the Confraternal order of Saint Hubert
Emprise de l'Escu vert à la Dame Blanche (Enterprise of the green shield with the white lady), founded by Jean Le Maingre dit Boucicaut and 12 knights in 1399 for the duration of 5 years
Emprise du Fer de Prisonnier (Enterprise of the Prisoner's Iron), founded by Jean de Bourbon and 16 knights in 1415 for the duration of 2 years
Emprise de la gueule de dragon (Enterprise of the Dragon's Mouth), founded by Jean comte de Foix in 1446 for 1 year.
In all cases, these orders were not centered around a nobleman - just a group of people vowing something to one another, and sometimes for a limited, set amount of time. Sounds a bit like an adventuring party to me. In the past, I have compared adventuring parties to merchant companies, which had a charter (i.e. "The Company of the Red Dragon shall plunder the red dragon's lair and split the proceeds as follows ...), but fraternal and votive orders could fit the bill as well.
To use the Emprise de l'Escu vert a la Dame Blanche as an example - it was founded for the protection of women suffering oppression, especially widows. Any woman so beset could petition the order, and they would send a knight forthwith to fight her oppressor personally.
The Emprise du dragon rouge, for example, could be founded by 6 knights (the adventurers) and their 12 retainers for the period of 1 year with the vow to hunt down and kill the red dragon Aglemire (and plunder his lair, of course).
Even better, an order founded by Lawful (or Good) characters could be a great cornerstone for a campaign. A band of adventurers could, for example, make it known far and wide that their order could be called upon by all honest folk who are being oppressed by foul wizardry, and then sit back and wait for the campaign hooks to roll in. They'd be a little bit like a medieval A-Team.