President Biden's most far-reaching mandates use four mechanisms: Federal employees must be vaccinated, contractors to the Federal government must impose mandates on their employees, businesses with more than 100 employees must impose mandates under an order (not yet written) from the Occupational Health and Safety Administration to insure workplace safety, and employers who accept Medicare and Medicaid payments (both from the Federal agency, CMS) must impose mandates on their employees.
Note that none of the mandates apply to state governments/ There may be a possible exception of state agencies that have Federal contracts, eg, universities, health departments, transportation departments, and others.
These mandates are not universal, they are limited in specific ways and appear to adhere to and fulfil Federal laws. Three of these mandates are imposed on Federal employees and contractors with the federal government. Not much there for a state to hang a lawsuit on. States can't interfere with Federal employment practices or contractor policies. Worker safety is a well-established Federal responsibility with laws tested in court. Not much opportunity for a state to interfere there, either.
So no, there doesn't seem any way for a state governor or attorney general to do anything but bluster for their constituency. But still, give them a chance to show up in court and see if their claims pass the laugh test. Then we can all have a good laugh.
Added Sept 15, 2021
There might be some room for some employers to dispute the mandate in court, based at least partly on the notion that spreading a virus is different from toxic chemicals. Any OSHA mandate is likely to allow the alternative of wearing a mask and getting regular, frequent testing, to maintain the safety of all workers. There is likely to be litigation but there is no logical reason for a state to be involved.