Care for disabled people in the Latin Middle Ages, whether from a congenital disability or one acquired later in life, fell to the person's family in nearly all cases. There are a few special, rare cirumstances where another situation might arise. Such as:
In the high and later Middle Ages, monasteries and then city convents might operate a "hospital" as holy charity. In medieval Europe, this would be more like a modern hospice or homeless shelter--a medium/long-term residence for the indigent sick and disabled.
For guild workers injured on the job, there are a few (very few) surviving guild regulations from the late Middle Ages that describe payments rendered to members who could no longer work
Especially in cities, people with disabilities seem to have been over-represented among the begging population
Late medieval England established a system of "guardianship" for aristocrats/other propertied people with cognitive disabilities--well, people judged in the eyes of a court to be unable to comprehend how to manage their money and property. These people would be considered wards of the king by law, but their guardianship--responsibility for the person AND control over their property--would generally be delegated to a favored noble. As you can imagine, this resulted in some bitterly contested court cases.
In a few incredibly rare, incredibly privileged cases, medieval (almost entirely) women used their chronic illnesses or other disabilities, in conjunction with a super-saintly life, generally a claim to divine religious revelations, and the support of male clerics to help make a case for "living sainthood." Some of these women, like Margaretha Ebner in 14C Germany, were already nuns. Others, like Margaret the Lame of Magdeburg (Margareta contracta), were supported by clerics/the Church in becoming urban recluses. They would spend the rest of their lives in a small "cell" attached to a church, often with one window into the church to observe Mass and one window to receive (for a very limited portion of time each day) visiting pilgrims who sought spiritual advice from someone who spent the rest of their day praying.
I have a few previous answers on medieval disability history, if you're interested: