The most common and most likely answer to this crossword clue is the 13 letter word DISAPPEARANCE
I've seen this crossword clue in The Times
D
I
S
A
P
P
E
A
R
A
N
C
E
Buddy explains!

The straight part
We know that "vanishing act" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case vanishing act = disappearance.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 DISAPPEARANCE mean?
DISAPPEARANCE noun- The action of disappearing or vanishing.