The City of Austin does not service private traps. It is also against the law to transport a rabies-vectored species without proper licensing (coyotes, raccoons, foxes, skunks, bats). It is best to contact a pest control company and to figure out the root cause as to why wildlife is coming around.
City of Austin Ordinance § 3-2-4 - Hunting and Trapping Wild Animals:
Except as provided in Subsection (B), a person may not knowingly shoot, kill, or hunt a wild animal; or use a steel-jawed spring trap or any other type of trap that could injure a trapped animal or person.