On our last visit someone in our group took a second drink to-go at Oga's so I think it can be done. There is one drink that has some visual effects with dry ice ... that one I don't imagine you could take to go. And, yes, you can walk around the park with a drink (alcoholic or non) - although be aware that Disney just updated their mask rules and you must be in a secluded area/socially distanced to remove your mask and take a sip or bite of anything. So, at least for now, there is no walking along and sipping/snacking as you go.
But you do need a reservation just to get inside (at least most of the time). Oga's is very, very popular. And there is no specific walk-up/take-out area. For us it's the atmosphere that we're going for (we're Star Wars fans). I don't think we'd bother to make a reservation and special trip to take one of the drinks to-go. In Hollywood Studios I'd be more inclined to get a to-go drink from the Brown Derby Lounge or Tune In Lounge or Baseline Tap House.