Inidividuals will band together and run services and businesses on a case by case basis.
Anarcho-communism... is a political philosophy and anarchist school of
thought which advocates the abolition of the state, capitalism, wage
labour and private property (while retaining respect for personal
property, along with collectively-owned items, goods and services) in
favor of common ownership of the means of production, direct democracy
(among communes, participatory democracy), cooperativism, equal
distribution of valuables, and a horizontal network of workers'
councils with production and consumption based on the guiding
principle: "From each according to his ability, to each according to
Without a state or central government to oversee, individuals group together and work together to preserve and maintain what they value. In essence, everything is a cooperative. Public services can work this way - the National Guard of the US is composed entirely of volunteers, and fire departments and ambulance services are run by volunteers in some places. The police force of your world can be a combination of volunteers and prior officers who organize themselves and are supported by the communities they protect.
Your stores and services the survive do so because they are valued by individuals, who group together to keep these things going. The anarchic aspect means that each business entity and service that survives the apocalypse might have a different story behind how it survived - one might have been taken over by the workers who run it, one might have been resuscitated by customers who miss it, one might be claimed by a religious organization, and so on.
Some of these things will have the same name, but be very different from how they were before the fall. It will be fun to write and fun to read!