What probability distribution would be more appropriate for monthly rate of going to the store?
I think you were right with your initial consideration of Poisson. I think that you can assume that the frequency with which people go shopping for a given good can be considered independent. Of course this assumption might not be perfect (like maybe you could argue that Person A buys item X every 3 days. However, if there is enough people I don't think this should matter too much).  Overall, sometimes I think if you need to use a distribution to approximate a process, you need to pick the best option and it seems like Poisson is best here.
Also, It can't be normal since its bounded (ie the lowest number of people that can go to a store is 0)