Update:
I can see the reaction completing as follows:
NH3 +
H2O 
NH4+ +
OH-CH3COOH

CH3COO- +
H+H+ +
OH-
H2O______________________________________
NH3 + CH3COOH

NH4+ + CH3COO-
This is an explanation that makes more sense to me. As water molecules are formed between the OH- ions (which come from the reaction between NH3 and water) and H+ ions, more and more NH3 and CH3COOH are ionized until completion. However my textbook explains completion with this reaction:
H+(aq) + NH3(aq)

NH4+(aq)
...essentially claiming that "the H+ ions from the CH3COOH readily combine with NH3 molecules to form NH4+ ions."
So, who is right? In other words, does NH3 really accept H+ ions directly? Or does NH3 react with water first, so that the hydroxide ion produced (and NOT NH3) reacts with the H+?