The most common and most likely answer to this crossword clue is the 4 plus 4 letter word TEAR DUCT
I've seen this crossword clue in Independent
T
E
A
R
D
U
C
T
Buddy explains!
The straight part
We know that "supplier of eye drops" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case supplier of eye drops = tear duct.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.