I managed to get through the first days at relative ease. After several attempts, I can't earn enough money on the day I'm at to feed and medicate my family. Is a stalemate situation possible?
First of all, it is entirely possible to play the game without making any mistakes. See this video (I didn't watch it, but it claims to be a perfect playthrough).
As Matthias noted in a comment, you are not expected to be able to always play from the latest day and get further. There are some situations where because of a past action, you will be arrested, or your family will die, or some other ending. You are encouraged to continue from a previous day or begin a new game and attempt to reach a different ending.
It is entirely practical for a skilled player to correctly process 15 applicants in a day (or 17 and 2 citations), earning the 75 credits required to feed 5 family members and rent a class 5 apartment.
Though this is an interesting question- is a softlock possible? For purposes of the question, this means a setup which will always end in ending 1 or 2.
Days with scripted attacks end after a specified number of migrants. So is it possible for there not to be enough, with zero savings and a single, very sick son?
If the inspector is in a class 8 apartment, he only needs 30 credits- 25 for rent and 5 to treat his son. No terrorist event allows fewer than six applicants. Additionally, the inspector can get 20 credits for shooting a terrorist.
If the inspector is in a better apartment, moving back to class 8 will become available earning 75 credits.