​Does the Order apply to Local Education Agency staff that don't work at a school site?  For example, employees that work at the school bus yard or district staff that work at the central office?

Yes. The Order applies to all school workers, without any explicit limitation based on assigned job sites. 

The Order defines "worker" as referring to all paid and unpaid adults serving in the following facilities: public and private schools serving students in transitional kindergarten through grade 12, except that it does not apply to home schools. Further, it does not apply to childcare or to higher education.

Covered workers include, but are not limited to, certificated and classified staff, analogous staff working in private school settings, and volunteers who are on-site at a school campus supporting school functions.

Does the Order apply to state preschool programs or other early learning and childcare programs if they are located on K-12 campuses? 

No.  As stated in the Order, the Order does not apply to childcare settings, which include state preschools.  While CDPH encourages all workers to be vaccinated to protect against COVID-19, workers who serve exclusively in such childcare settings, regardless of location, are not covered by the Order.