From a federal income tax perspective, there is clearly no prohibition.  Universities and other charities regularly hold equity interests in for-profit entities.   It is the basis of their “endowment.” 

Depending on the structure of the new entity, however, it could be considered a joint venture by the IRS and, if not controlled by the University, could generate unrelated business taxable income.  (See Ready Reference Page: “IRS Says Charities Must Control Joint Ventures”). You will also want to be sure that you will not be providing a private benefit to outsiders that could affect your basic exemption, although that seems unlikely in a situation like this.  And you will want to determine whether rental to a for-profit business, even one partially owned by the University, will cause some or all of your STEM building to be subject to local real estate tax.

Assuming you answer each of these questions (and others) to your satisfaction, that does not mean that you should automatically jump at the opportunity.  It could open unique opportunities with the new company, but it might cut off other opportunities with competitors.  It could significantly reduce contributions from competitors.  It could raise conflicts with your other owners if you don’t want to operate the company for maximum profit, which is to some extent inconsistent with using it as a significant training facility. 

I am sure that others have made, or at least considered, similar arrangements.  I suggest that you use your broader university network connections to get practical points of view and experience on whether it ultimately makes sense to proceed.