Greet us excitedly, giving wave, perhaps (7)
I believe the answer is:
gesture
'wave perhaps' is the definition.
(waving is a kind of gesturing)
'greet us excitedly' is the wordplay.
'excitedly' indicates an anagram (letters excited or stimulated into a new form).
'greet'+'us'='greetus'
'greetus' is an anagram of 'GESTURE'.
'giving' acts as a link.
(Other definitions for gesture that I've seen before include "Arm signal" , "Sign with the hand" , "Movement, indication" , "wave, perhaps?" , "Action done for show" .)