The most common and most likely answer to this crossword clue is the 8 letter word AXILLARY
I've seen this crossword clue in The Times
A
X
I
L
L
A
R
Y
Buddy explains!

The straight part
We know that "of the armpit" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case of the armpit = axillary.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 AXILLARY mean?
AXILLARY noun- Any of the feathers connecting the undersurface of the wing and the body, and concealed by the closed wing.
- Of or pertaining to the axilla or armpit.
- Situated in, or rising from, an axil; relating to an axil.