The issues to address
1 - supply chain. You have central distribution centers that receive incoming goods in mass bulk. These goods are then distributed to stores and other sales destinations by trucks. The first thing a city would have to address is how to make a valid distribution system without automotives doing the last leg. Light rail would be a bit difficult as you'd need a rail way to each and every store loading bay really. Is electric trucks an option (assuming the electricity was produced by something other than coal?).
2 - Purchased goods. I just bought a fridge and stove for my home...how exactly do I get my appliances delivered to my home without a vehicle? I'm decently physically fit and could probably push a cart with it (getting it onto a LRT would be great fun)...but the majority of the population would struggle with it. GO one step further...how do we get the construction equipment (supplies and construction vehicles) to the site of a new home?
3 - Garbage. We got the goods to where they get consumed...now how does the inevitable trash get taken away?
Many modern day cities actually do a fantastic job of keeping this supply chain hidden from the populace...done on off hours or early morning and as seemless as possible. One of those functions that you barely notice happening until it fails and you realize how dependent on these functions we are.
Certain cities are more ready for this than others. Calgary Canada is an amazing sprawling city designed well after the dominance of automotives and the road network is designed to help cover this massive area. Ghent in Belgium saw it's rise in early medieval times and could probably go through with a vehicle ban and have little effected in terms of day to day life...they actually have a three tiered transit system of the train for intercity, railcars for common transit routes in city, and buses to service the outlying routes...most North American cities have two tiers of rail + bus. Remember this policy will have a huge impact to cities designed in medieval times prior to the car as opposed to cities designed well after the cars dominance in personal transportation.
Just one to add as difficulties in this:
The rich! Don't underestimate the ability to influence policies with money...if people with money wanted to drive on the streets of a city at the expense of every other persons health, they will. Very few politicians would risk losing their parties funding over something that angers their funding base...if they did political parties that oppose this vehicle ban would suddenly find themselves swimming in cash.
Something would have to be done to address the distribution system. I'm actually not so sure on solutions here...an online order system that drops off your purchases at a distribution point where you come to to pick them up would be a possibility (decentralization of purchasing goods and potential elimination of stores), but even then we are facing the challenge of getting your goods back to your home from there. Whatever the solution, the city would have to implement this prior to a ban...even if the implementation includes a phased approach where personal vehicles are gone first while the distribution network remains until a suitable solution to replace it is found.
Public use cars - I'm not sure if you are aware of the 'car2go' program where smart cars are available for people to use as required and then leave on the street (or drop off point) for the next person to use. It's a potential 'phase out' approach to allow people to move their goods as needed.
Streets become obsolete to some degree...why would a city have roads that are no longer used? I'd imagine emergency vehicle use of roadways would still be required in any event. The freed up land can be re-used for either a transit system or whatever replaces the distribution network. In urban centers, streets become open to all pedestrians to walk. In Suburban areas, urban farming or other such ideas could also be used to cover the freed land as well. I wonder how many homes will suddenly find themselves with a nearly useless attached garage?
I can't imagine any of these changes occuring in less than a decade as a combination of city planning and public consultation (yay bureaucracy) along with construction would be a drawn out process...and scarcity of material may become an issue if all cities attempted this at once. It'd be a massive boom to the manufacturing and construction industries at any rate.
Would car ownership outside of cities be allowed? If I owned a car, but lived where it was not allowed to be, would I have to park it outside of the city and come inside from there? Heh, it'd create an interesting phenomenon of giant parking lots surrounding cities at train stops to take people back and forth from their vehicles to their homes.
There might be a population density rearrange...very light (outside of urban and can use cars) would become far more attractive, or extremely high density (easy to be included in the supply chain) would be cheaper to maintain. It may result in the death of the suburban family white picket fence dream