A few thoughts:
When I read this, I hear "I'm going to charge people to remove problem pests". This won't require any hunting/trapping license, but it WILL require a business license, and will require you to hold liability insurance. This will be cost prohibitive for you, unless you do it on a cash basis illegally. Your overhead will be higher and your method less effective (and more costly) than comparable services that are readily available: local animal control is offered through the city/county/state, and pest control by licensed/insured professionals are readily available as well. Your competition, with your limited resources and service offerings, will crush you.
On the other hand, if you're referring to trapping and selling the hides for money, rodents/rats aren't going to cut it. If this is the case, you'll need all appropriate hunting/trapping licenses for your state, and will need to follow your states designated game seasons. Depending on the size of your city, you'll also likely run into problems with municipal codes that prohibit hunting/trapping within city limits. You will ALSO be at risk for liability if you would accidentally catch/injure/poison/kill someone's pet. Break your neighbors lap-dog's leg and you'll likely face a lawsuit, making it risk-prohibitive to pursue. The resources required to sustain (i.e. pay for itself) a trap line in the city is cost-prohibitive. Your volume (animals caught per trap per day) will be incredibly low based on the population, so you'd be much better off to DRIVE OUT OF TOWN and set trap-lines in the country if fur-selling is something you think you'll be able to make money at.
Personally, I wouldn't mess with it. Way too many weeds to get hung up on through that process legally, regardless of what your intent might be.