How does Vianne describe the connection between food and family, friends, or community?

Although about half of the book is told in the 1st person
from Vianne's perspective - she is not a particularly introspective character.  In fact,
one of the beauties of her personality is that she tends to tell the story as it is
happening, she does not make any apologies for her thoughts or actions, but generally
speaking, she remains unbiased towards others and circumstances.  In this way, to answer
your question, Vianne never makes a direct connection between family, friends and
community.

In order to answer what is
Vianne's connection between these things, we must only look at her actions.  Her version
of events is straightforward and sounds as honest and unimbelished as if she were a mere
observer, rather than a participant.

Vianne uses her gift
of cooking (not just the huge feast for Armande's birthday, but every delicacy in her
Chocolaterie) to do two things.  One, to provide the kind of
comfort necessary for people to want to come to the shop and talk. 
She does most (if not all) of her relationship building over something sweet to eat or
drink.  She can "guess" people's "favorite" treat - and uses that charm to coax them out
of themselves.  Characters in the Chocolaterie find themselves opening up and sharing
things about their past, their secrets (hopes as well as fears), and their sorrows. 
Vianne is always willing to listen.

Second, she uses food
to bring people together.  Armande's party is only one example of
this.  Simply put, once a character has opened up (emotionally) to Vianne, she finds
someone else in the town and makes a connection.  She does not selfishly get close to
everyone and keep them all to herself.  Instead, she allows them to trust her - then she
helps them to make a similar connection with someone else.  In this way she connects
Armande to her grandson, Luc.  She connects Luc with his mother.  She helps Josephine
leave her husband - and in so doing, Josephine is able to make friendships with several
people whom she seemed afraid of before - including
Roux.

Vianne's gifts are cooking and connecting.  Through
these she uses simple pleasures to bring people together.