The most common and most likely answer to this crossword clue is the 9 plus 4 letter word UMBILICAL CORD
I've seen this crossword clue in The Times
U
M
B
I
L
I
C
A
L
C
O
R
D
More clues leading to the result UMBILICAL CORD
Buddy explains!
The straight part
We know that "babyfood supplier" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case babyfood supplier = umbilical cord.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.