Well, I guess that you have region restriction enabled by shipping zone, and once the user gets into your site, the geoprompt appears and he/she puts the postal code. In this way, just the user just will be able to see the inventories set on that shipping zone.
On another site, if you don't enable the region restriction mode, every inventory (store) will be displayed to all users, so they will have to choose the store that belongs to their city, but you won't be able to avoid that they choose another inventory... So I'm not sure if it's possible to do exactly what you want, but maybe trying this second option is the most accurate.
Please, tell me if it helps,