Sure, between fighting off starvation and the roaming bands of starving people fleeing the cities and raiding your farm lands........
keep in mind that you no longer have electrical power or refrigeration............and that you've quite likely already rid your neighboring forest of all squirrels and deer and possum and whatever edible animal that used to wander there.
Seriously, do you just make crap up?
What roving bands? To get to this town would take days and days of walking from Cape Girardeau and maybe 3 days from Poplar Bluff. Those are the “big cities” around. These “roving bands” would need to be carrying all the food and water they need.
That’s a ludicrous idea.
Do you know what grows on farms? Farm animals. Pigs and cows and chickens. Some farmers keep a stocked pond because they like catfish, and blue gill are common in the ponds to keep mosquitos down. So while squirrel and deer would likely suppliment the towns meat supplies (as it does in normal times), the town wouldn’t be reliant on it.
Penicillin can be made using white bread and culture dishes. (Most high school biology classes have a whole series of petri dishes) Once you get a culture of penicillin growing, you can keep it going.