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This will involve filing an application for re-zoning with the city's planning and zoning commission. Contact a representative in that office and he/she will provide you with the appropriate forms to fill out.
Once your application is complete, the P&Z commission will perform an informal investigation regarding whether the change is positive or negative, or otherwise appropriate. These findings will be presented to the city council or board, and they will vote on whether or not the change should be allowed.
If the P&Z are on your side, it will be a slam dunk! If it opposes the re-zoning, you're going to have a tough time convincing the board/council to overrule the P&Z officials.
Anything you can provide that will help convince the P&Z commission to change the zoning is helpful.
I would ask the P&Z what would help your chances before spending any money - it may help if you can address the supply issue and whether or not a trailer would be better than using your vehicle. However, if you're selling on a public street corner, you may can just get a permit and not have to worry about zoning.
Your best bet is to "buddy-up" with the P&Z folks and ask them what would be the best way (under the city ordinances and laws) to get your business all of the permits it needs to operate legally.
Good Luck! Now, I'm going to get a hot dog!