No, DataCamp for Classrooms is made accessible for free with the express intent of teaching instructors and students. Just teaching staff are eligible to apply, and they are limited to inviting only their students and teaching assistants.
Other teachers in the school should sign up for a DataCamp for Classrooms account as well. In the DataCamp for Classrooms account, teachers/professors are not able to invite other non-teaching members of the school staff.
If your non-teaching faculty is interested in using DataCamp for Classrooms for their own purposes (such as attendance and budget tracking/research), have them contact Data Camp Sales team to purchase an Enterprise account, which is designed for very large organizations like schools or entire school districts.