The most common and most likely answer to this crossword clue is the 6 letter word ARCHER
I've seen this crossword clue in The Times
A
R
C
H
E
R
More clues leading to the result ARCHER
- A Catholic, the female who may have a quiver?
- Endless stuff about female Olympic competitor?
- Comparatively roguish sportsperson aiming for the gold?
- Old soldier often looking for a quarrel
- Person walking miles leaves old soldier
- Arrives to entertain revolutionary old jockey
- Demonstrator heading off one with a bow
- More playful Ambridge resident
- I shoot Bow revolutionary in a railcar on vacation
- Arrives to entertain revolutionary bowman
Buddy explains!

The straight part
We know that "who may have a quiver" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case who may have a quiver = archer.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 ARCHER mean?
ARCHER noun- One who shoots an arrow from a bow or a bolt from a crossbow.
- The bishop in chess.
- Knowing, clever, mischievous.
- Principal; primary.