Seems like the shelter can indeed be the source of the problem.
It is common wherever cats are housed in groups
and it is readily transmitted between cats.
Transmission is typically by contact with infected feces. This means that the litter box is the usual place where infection takes place. This infection is unusual in cats that free-roam outdoors (no litter box) or who live in homes where there is only one cat. The virus enters the new hosts body via the nose and mouth.
Households with fewer than 5 cats eventually spontaneously clear of coronavirus. Households with more than 5 cats virtually never clear of coronavirus.