My guess is business schools tend to have a full-time staff to make decisions like that, perhaps chaired by a oversight committee composed of a dean and some professors.
For doctoral programs, academics themselves make a lot of those decisions, because they are training their peers. But in professional schools, the professors are probably not highly involved in such a routinized process (just like most undergraduate admissions).
That doesn’t mean that a professor’s recommendation can’t carry weight. If a professor wants you to work with them on something, an admissions committee would usually take that fairly seriously.
Final admissions decisions at business schools are generally made by professional admissions staff, not professors. However, a recommendation from faculty of that school would probably not hurt your chances.
You should think about how you would approach faculty by volunteering at school events, as your intentions would most likely be pretty obvious. It might make more sense to just volunteer for organizations or causes that you actually care about and that would add to your overall application in a consistent way.
Honestly, you’re not going to be the first guy that’s tried to do this. They’re going to smell you from a mile away, and chances are they don’t like this approach. Being genuinely interested in their program and being a good fit is a better way to go.
I’m an alumni interviewer for my school. I can tell you that the faculty is rarely involved in MBA admissions decisions. There is an admissions committee compromised of dedicated admissions staff, some of whom were alumni of that school. In addition, alumni also help with interviewing so have some say in the admissions process as well.
Volunteering at business school-sponsored events like MBA fairs will add zero value to your candidacy.
Now, here’s something that schools don’t want you to know but it happens all the time: there are certain schools where faculty members and alumni can write a recommendation on behalf of a particular candidate. There is no official weighting to this, but it does happen and it does go in the file. Some schools will tell you NOT to submit additional materials or to seek extra recommendations, and if a school abjectly discourages it, then you probably shouldn’t do so. But, there are other schools like Chicago-Booth where current students and faculty are allowed and even encouraged to “recommend” people, and there are schools ranging from undergrad to Ph.D. where faculty can submit an informal letter about a student if they feel very strongly.
Some here might argue that this is nepotism but connections always matter, as they have since the beginning of humanity as far as I can speculate. It’s just that sometimes they matter more than other times.
I used to work at the MBA admission office, though i can’t say for every business school, but mine has a full time committee (i think 6 people?) completely separate from faculty staff (so no dean or professors in the mix).