If I use Weavile's "Vilify" attack to discard one or more Exeggcute, can I use Exeggcute's "Propagation" Ability to immediately retrieve it back into my hand?
No, you cannot. Propagation can only be used on your turn before your attack, not during your attack nor afterward.
Source: Plasma Freeze FAQ (2013-05-16), TPCi Rules Team (2013-05-16)

Relevant cards

Note: All applicable cards may not be listed below.
BW10_102
BW9_4