Do you need a permit to start a lost pet searching job?

In the US, business licences are usually regulated by local governments, either your city government or county government if you live in an unicorporated area. The regulations vary tremendously from one jurisdiction to another. Some jurisdictions do not require business licenses at all. Most local jurisdictions do require them but many make exceptions for casual businesses such as baby and pet sitting especially when conducted by minors. The only way to find out if you need a license for any particular work, in any particular location is to call your local government and find out.