As previously stated, it depends on where you live. Despite popular belief, this isn't that easy of a topic to address.
Currently, in the US, there are more than 24 states that have some sort of requirement for nuisance wildlife damage control (off the top of my head there is CA, CO, CT, FL, GA, IN, IL, KY, MA, MD, MI, MN, NC, NY, OH, OR, PA, RI, SC, TN, VA, WA, WI, WV, DC). Some of these states have an actual license and testing requirement (CT, IL, IN, KY, OH, GA, MA etc.) while others require a hunting and/or trapping license or special permit. For some states you will first have to register as a business with the appropriate Secretary of State and Tax department, while in others they don't check that requirement. Additionally, there are states where you have to write about your experiences with controlling wildlife and submit it to the regulatory officials or do a verbal interview in order to determine if you have enough qualifications to be given a permit/license. Some states will require the license/permit/authorization regardless if a fee is being charged or not, so don't think that because you'd be doing this for free that you don't need to check first. I don't know that current status, but I do know that Canada is working on a wildlife control program that would be recognized in all Provinces and Territories.
Realize that even if you are granted a license/permit/permission it doesn't mean that you can use air rifles. More states are going with the operator having to submit a review of the conflict situation and list non-lethal control methods that must be tried before any lethal control methods may be used. If you are able to use lethal control methods, start with looking at the regulatory laws to determine if you are allowed air rifles. Even if air rifles are legal, you also need to see if there are any rules about using a suppressed or silenced air rifle on a wild animal. For example, because Ohio had a Revised Code rule that suppressors, mufflers or silencers could not be used when shooting a wild animal unless otherwise stated by the Ohio Division of Wildlife (DOW), we had to add a special clause in our nuisance wild animal Administrative Code to allow suppressors, mufflers or silencers in order to use air rifles that are so equipped (Revised Code trumps Administrative Code and because authority was given to DOW in Revised Code we are able to make the necessary change). This is kind of the issue that is currently going on in Michigan.
Additionally, you need to understand not only the state laws and regulations, but also Federal laws and based on where services are performed, municipality laws or township ordinance. This is generally where most problems occur. Quite a few municipalities and townships have laws and ordinances that prohibit "propelling an object at a wild animal with intent to hurt or kill." Other issues can be "discharge of an air rifle/firearm (you need to know how air rifles are classified for your service area) within X number of feet of a road or dwelling." And then you have those areas that have designated as "wildlife/bird sanctuaries" that need to be considered as well. Although this can look daunting, don't think it means you cannot use an air rifle. Rather, what it means is that you may have to apply for special permits or permission in order to use an air rifle to resolve a conflict situation.
To find out what is required for a state, go to the regulatory authority website and look for nuisance wildlife or nuisance wild animal rules (some states have Dept of Fish & Wildlife while others will have a Dept of Environmental Conservation or Dept of Wildlife). You can also contact a wildlife officer, but realize that they may not know all of the rules and licenses associated with doing nuisance wildlife or nuisance wild animal control so it really is up to you to do the research to protect yourself.
Eric