Big River Grille is an operating participant. You might be able to call them directly and reserve a table, but I'm not sure about that.
ESPN is Disney-owned and operated. They don't take PS because it would be a logistical nightmare! You never know how long people are going to stay (If their game is on, and it's good, sometimes they stay through the whole thing, and then stay and buy beers to celebrate!), and especially during football season it's impossible to predict who's going to come in when. I remember one guy who came in every week for Monday Night Football -- he got there at 3 in the afternoon!
Usually both restaurants are walk-ins or something close to it. ESPN is divided up into two parts: Sidelines, the bar area (which is full-menu and full service) where you seat yourself, and Sports Central, where you'll wait in line for the host/hostess to seat you.