using alton brown's aged eggnog recipe
You probably can, but why would you want to? When eggnog is homemade and delicious, it deserves a beautiful punch bowl on the center of the (or it's own small) table. Imho. A gallon dispenser doesn't sound very festive.
I think a dispenser would clog. I agree with Chef June, a punch bowl is best. A pitcher could work too. If you want to use your dispenser, stick to something clear.
Agree with answers so far. And if you don't have or don't want to buy a punch bowl (for a single use), get a large glass bowl that you could use year round for other purposes in the kitchen. Then place some nice greens or a red cloth around it on a buffet table.