The most common and most likely answer to this crossword clue is the 13 letter word NEIGHBOURHOOD
I've seen this crossword clue in The Sunday Telegraph
N
E
I
G
H
B
O
U
R
H
O
O
D
More clues leading to the result NEIGHBOURHOOD
Buddy explains!

The straight part
We know that "vicinity" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case vicinity = neighbourhood.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 NEIGHBOURHOOD mean?
NEIGHBOURHOOD noun- The quality of being a neighbor, of living nearby, next to each-other; proximity.
- Close proximity, nearby area; particularly, close proximity to one's home.
- The inhabitants of a residential area.
- A formal or informal division of a municipality or region.
- An approximate amount.
- The quality of physical proximity.
- The disposition becoming a neighbor; neighborly kindness or good will.
- Within a topological space:
- Within a metric space:
- The infinitesimal open set of all points that may be reached directly from a given point.
- The set of all the vertices adjacent to a given vertex.