The most common and most likely answer to this crossword clue is the 6 letter word DOGNAP
D
O
G
N
A
P
Buddy explains!
The straight part
We know that "Grab someone's pet" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case grab someone's pet = dognap.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 DOGNAP mean?
DOGNAP verb- To abduct (a dog).