Institution without kosher food:[1] With respect to placing an individual in an institution for the mentally impaired that does not provide kosher food, the Poskim rule that it is permitted to send a child with severe intellectual disability to a non‑kosher facility.[2] This leniency applies only where the individual is deemed completely incompetent (shoteh gamur). However, if the individual is not fully incompetent and possesses some capacity for understanding and education in matters of Torah and mitzvot—such as individuals with Down syndrome or similar conditions, or a cheresh (deaf individual) who is capable of speech and comprehension—then one should refrain from sending them to such institutions, as they are amenable to at least partial religious instruction and observance.
[1] Chasam Sofer O.C. 83 and in Hagahos 343:11; Kaf Hachaim 343:22; Igros Moshe 2:88; Shevet Halevi 9:4; Minchas Asher 2:47; Piskeiy Teshuvos 343:10
[2] The rationale is that this is considered indirect consumption of non‑kosher food rather than direct feeding, which is prohibited.
