Hi Eladt -- first year MBA student at Anderson here, I can touch a bit on my personal experience. My fellowships are funded both through the Forte Foundation and an individual donor. There wasn't a separate application for these, just my application for admission, and I wasn't involved with the Forte Foundation before applying. I agree that having an overall solid story and credentials increases your chances at a fellowship, but I'm not sure it's entirely necessary to hire a consultant (I didn't). That said, I would only apply to schools if you feel fully confident in your application, so hiring a consultant can be helpful in that regard.
Additionally, there are some fellowships that students apply for during the beginning of their first year. For example, the Fink Center sponsors a variety of fellowships in banking, private equity, and asset management. The financial awards associated with these fellowships are nominal, but they offer designated mentor relationships (in addition to being named a fellow and included on the website -- helpful for recruiting).
Hope this helps!