It is possible that the university has a section or even "subsidiary company" through which academic staff is "outsourced" as consultants to industry. What I just wrote looks horrible, so I will try to elaborate.
There might be a conflict of interest issue when an academic works for both a university and a private company, depending on the situation. That can be overcome in legal ways via separate companies etc, and I am not experienced enough to delve on that issue. Suffice to say, it happens and in ways that are the whole package: right, moral and legal.
In addition, some universities have a specialised agency (or provide the opportunity) to liaise academic staff to companies. They act as the intermediary, negotiate terms with the company, undertake the contract, arrange for payment and terms etc, and therefore there is no violation of some kind. In essence the university arranges for a fractional or consulting contract and centralises the process. I do not know enough to provide details, but the University of Oxford comes to mind. This is an option worth exploring and means a company will be negotiating with the university instead of a private individual.
I will not go into the moral aspects of academia - industry relations, but I see nothing wrong on the outline of the cooperation the OP suggests. It is something fruitful for both sides, and most academics would at least consider it. The obvious option is to email a spacific person, if the profile fits, and a to-the-point factual but not overly detailed email that intends to lead to a face-to-face discussion sounds fine. Another avenue is the Head of Department and/ or administrative contact, which might be difficult to locate for the uninitiated to the supreme mysteries of academia. There is typically a point of contact on the Staff list, and if the OP is looking for a list of people that is also worth considering. It is situation specific, however - some departments/ people are slothful, while others hyperactive.