The most common and most likely answer to this crossword clue is the 8 letter word STRONGLY
I've seen this crossword clue in Financial Times
S
T
R
O
N
G
L
Y
More clues leading to the result STRONGLY
- Violent sort, no good, starts to lambast you in forceful manner
- Scheming to kettle mob hotel ejected with force
- The way Republican just drinks gallons to a large degree
- Unjustly, first off, going after saint forcefully
- Assertively but inappropriately, holy man replaces wife
- I left Tony cavorting with girls vigorously
- Using force in street unacceptably after wife leaves
- Good man mistakenly losing head with force
- Mightily in error heading off to follow holy woman
Buddy explains!

The straight part
We know that "vigorously" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case vigorously = strongly.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 STRONGLY mean?
STRONGLY adverb- In a strong or powerful manner.
- Very much.