Not sure a good solution exists. There is more demand than slots. Waiting a month for groceries while stock changes and you are left with 50% of your original list doesn't work either. If he can get a slot or not it is better to know immediately so he can shop around.
This service isn't for your father, another service that provides help for people with parkijsons would assist him in these and other situations.
This service absolutely is for people like my father and people who are not like my father. A shitty user experience that is totally solvable is the problem. He's also immunocompromised with congestive heart failure and high blood pressure. Due to this terrible user experience, he's forced to leave his home to go to the supermarket in person when he's one of the people that should not be leaving his home.
A good solution is the FIFO queuing system that Ticketmaster uses for ticket sales or the one that Yelp uses for first-come-first-serve restaurants like Nojo in Hayes Valley, SF.
Let people sign up for slots, put their name on a list, and prompt them to choose slots as they come up. This can also be used to prevent people from getting two slows back to back before someone else gets their shot if you make it so they can only sign up again after their name has been called and they've gotten a slot.
This service isn't for your father, another service that provides help for people with parkijsons would assist him in these and other situations.