Hospitallers were probably providing armed pilgrim escorts reasonably quickly after gaining autonomy in 1113. However, they only became fully militarised under the leadership of their second master, Raymond du Puy - probably around 1125.
For completeness, the Templars were active from c.1120 - albeit initially in small numbers.
I would steer away, however, from the idea of groups of brethren taking up arms on their own initiative in hierarchical, monastic orders. The consequences of engaging in violence on one’s soul was a problematic matter in the High Middle Ages, and not something entered into lightly by professed religious. Protecting pilgrims, i.e. one’s Christian brothers, was an acceptable option, but spilling blood would still have required ritual cleansing. Doing so without the support of their leadership risked lengthy and demanding penance and probable expulsion from the order.