Loosely translated, the paragraph means: "The man spends more money on liquor than on food for his children. His children are not fed enough. So landowners in the area do not mind letting the man shoot animals for food to feed the family."
In more detail:
A relief check is a tax return, welfare check, or other form of government assistance such as disaster relief (FEMA) or economic stimulus (IRS).
Green whiskey is the first pour of liquor from a batch of whiskey. It tends to be much stronger than later, clear pours. (Dismilis Canyon)
'Game' as mentioned in this paragraph is hunting game, or wildlife. In America at this time most regions allowed land owners to hunt the animals on their land with restrictions to prevent hunting animals during times in which they procreate and raise their young. Enforcement on those restrictions was often not strict unless the landowner specifically called the police (game warden) in to handle a problem.
"Of course he shouldn't" refers to Mr. Ewell hunting animals for food and spending money on liquor instead of spending his money on food directly. If drunk, his hunting skills are not likely to be good, so the children go hungry.
"take out your disapproval" is asking if Scout is going to get Mr. Ewell arrested for poaching (hunting game out of season), and as a result causing more hardship for the Ewell children.