Try calling your city or county animal control offices to locate free animal removal services, but be aware that you may have to hire a private company. Municipal animal control typically only provides limited removal services and may only handle domestic animals.
In some cases, your local animal control agency may be able to provide free traps for you to use, and then pick up the trap and relocate the animal once it is caught. In most cases, it does not send agents into people's homes to catch wild animals, and agents do not trap or try to catch the animals themselves. Loose or dangerous dogs are a notable exception. The agency also may be able to provide more help if the animal appears to be diseased, because rabies is a serious health concern.
To find affordable animal removal services, spend some time calling local companies. Ask them how they charge, since many charge by the animal and you may end up paying more than you expect if they catch multiple animals. Decide whether you want them to just take the major problem animal or do a thorough inspection of your home and yard to make sure there are no other animals hiding out. Try to find a company that specializes in the type of animal you have so that the trappers can work more efficiently.