About 30% of adults don't drink alcohol, so I would guess there are many who don't drink at Disneyworld. I've spent lots of time with recovering alcoholics and teetotalers alike (i.e., people who don't drink). I've never seen anyone judged for not drinking, I mean who would know if you are or not unless you happen to have a drink in your hand right then?
I'm pretty sure we had 0-1 alcoholic beverages for our trips in 2003 and 2009.
Our last trip to Universal in April, I had sparkling wine at the dessert party (DH did not) and we both had wine with our fancy steak dinner one night.
ETA: I didn't add my why. We aren't big drinkers in general, maybe once or twice a month. In 2003, alcohol would have been a budget buster. In 2009, we went with my in-laws who don't drink and we didn't want to offend them since we don't drink much anyhow, it was no sacrifice.
In general, as I said, we don't just always have a beer or wine with dinner. We don't go out every weekend. At a theme park, when dealing with heat and children and heavy food and expensive prices and lots of walking, it just isn't that "fun" to drink for us.