Chicago homeless migrants shelter faces backlash
Chicago homeless migrants shelter faces backlash

www.reuters.com
Chicago homeless migrants shelter faces backlash

Chicago homeless migrants shelter faces backlash
Chicago homeless migrants shelter faces backlash
* migrant's
(or my bad ? English isn't my first language)
It's correct how it is. It's acting as an adjective, as if you said "red shelter." "Migrant's shelter" would mean the shelter belongs to one migrant. "Migrants' shelter" would mean the shelter belongs to a group of migrants, which is also incorrect because they do not own it. Hope this helps :)
You don't have to have legal ownership of a thing to use the possessive. "Migrants' shelter" is fine, and I would say preferred.
Neither do you with that self-righteous attitude.