Are we that uncomfortable with human emotion that we have to mock and belittle a shed tear in a show that is brimming with emotion in spots? I would say that it is within the realm of probability that Dolly, Cornelius and Irene Molloy would all likely cry at some point in the show given the emotional roller coaster that Wilder and Herman/Stewart send them on.
I don't think they need to be sobbing so that it gets in the way of the singing, but I'd be shocked if they (or I) DIDN'T shed a tear at some point.
And- SPOILER ALERT- in the current production it is a lovely touch to give Vandergelder a couple of those moments as well.