Are there any places on the grounds to get drinks in-between restaurant openings?

Yes

There is the Mizner's Lounge on the second floor of the lobby, overlooking the lake. There are also the Beach Poolbar and a Courtyard poolbar. You're also within walking distance of the Polynesian Resort, which has the Tambu Lounge (lobby), the Sand Bar (pool), and one of my favorite bars on property, Trader Sam's Grog Grotto.

There is a bar on the second floor of the main building (where you check-in)

Gasparilla is always open and sells alcohol