The OSPI website provides much information on the steps involved in completing background checks. The website link lists the ESD locations and confirms that your teachers can utilize their local law enforcement agencies or private fingerprinting agencies. There is also a Q&A that answers additional questions.
For volunteers, there may be some different options: https://www.seattleschools.org/departments/volunteer/background-checks-for-volunteers/. They mention that statewide background checks are free to nonprofits and provide links to low-cost resources.