What penalties, if any, could be inflicted on the two companies for not removing the app from their app stores, and under what legal authority?
This is the order.
It cites as authority:
... the Constitution and the laws of the United States of America, including the International Emergency Economic Powers Act (50 U.S.C. 1701 et seq.) (IEEPA), the National Emergencies Act (50 U.S.C. 1601 et seq.), and section 301 of title 3, United States Code.
Looking at just the first of these, the International Emergency Economic Powers Act (50 U.S.C. 1701 et seq., the penalties are in s1705.
For a civil offence, they are fines of the greater of $250,000 or twice the value of the transaction. Note that this is per transaction so each and every download can attract a fine of up to $250,000 - 4,000 downloads is $1 billion.
If it is proved that this was done recklessly or deliberately (which leaving them on the store would be) then criminal sanctions apply; a $1,000,000 fine and 1 year in jail for any employee/executive/director with criminal responsibility per offence.
Would the companies be able to argue that they are not making transactions simply by not removing the app from the store?