The most common and most likely answer to this crossword clue is the 6 plus 4 letter word GARAGE SALE
I've seen this crossword clue in The Daily Telegraph
G
A
R
A
G
E
S
A
L
E
Buddy explains!
The straight part
We know that "local disposal" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case local disposal = garage sale.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.