Does anyone know if it's possible to simply pick up an appetizer like conch fritters at Olivia's "to go"? Especially in the event they can't take a walk up for drinks and fritters only?
For the most part, at WDW restaurants, no. There may be a couple that might allow takeout if asked, especially at resorts; sometimes guests report success at getting soup from Le Cellier, although just as often they seem to get turned down.
I am certain about Olivia's though. They do not do takeout at all. From what I understand they used to allow full takeout, but stopped several years ago because DDP guests were demanding that they should pay CS credits for takeout rather than TS credits.
There is a bar called the Gurgling Suitcase. You can order from Olivia's menu there. At my last visit, you had to eat the food in the bar; they would not package it so you could take it with you. You could definitely order some appetizers and eat them in the bar.