The most common and most likely answer to this crossword clue is the 12 letter word BACKSLAPPING
I've seen this crossword clue in The Times
B
A
C
K
S
L
A
P
P
I
N
G
Buddy explains!

The straight part
We know that "Vigorously hearty" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case vigorously hearty = backslapping.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does BACKSLAPPING mean?
BACKSLAPPING verb- To enthusiastically affirm or congratulate a person, especially by patting them on the back.
- Action of the verb to backslap