The support shop is the branch office. Presumably this is because the regular "shop" is the "shop" and anything that supports it is the local (branch) office.
If it helps, this word has the same meaning as 支局 which you will also be learning in this level.
This is a jukugo word, which usually means on'yomi readings from the kanji. If you know the readings of your kanji you'll know how to read this as well.
I go to the branch in that town every day.
My brother bought a used car at the branch.
My husband is the branch manager of the New York branch.