Does OSHA have regulations that would apply to employees in a higher education setting?

Unfortunately, OSHA doesn’t break out higher education establishments from the rest of “general industry.” So, covered universities would be covered by the same standards as those for manufacturers, retailers, etc. The OSHA standards are hazard-based or equipment/process-based, rather than industry-based, for the most part. 

For example, if you have a laboratory, then that could trigger some requirements, if you have certain specific chemicals, those could trigger requirements, if you operate forklifts or powered pallet jacks, workers would need training, if workers work at heights or are exposed to fall hazards, training would be needed. Other issues that might be common to your type of settings include: electrical, aerial lifts, sanitation, asbestos if doing renovation on buildings, emergency action plans, and fire safety.