An institution would likely use their broker's trading desk or a third party trading desk like BTIG. Those desks have access to a small list of large MM, a larger list of smaller MM and a group of hedge funds that focus on providing liquidity. They communicate over the phone and through a messenger service like Pivot, to "shop" the order. As a small market maker, up to 2010, I used AIM to respond to institutional option orders. I was required to respond with a 2 sided market for any request with a size of at least 100 up or I would be ignored. Most orders that came across were between 1000 and 10,000 and many paired with stock.
Bob