The church was hit hard. In the 14th century, there may have been a monarch, but in modern terms there were no government to dispense social programs. If you were poor, if you needed food, shelter, medical attention, or if your children needed education...the church was the only place to go for help. The Catholic church was building, and running, schools, hospitals, and shelters. Catholic monasteries continued to be the centre of learning in Europe. (If you aspired to be a scientist, the church was far and away the largest source of financial support for science and scientists, too.)
The plague hit hardest among the most numerous segment of society, the peasantry, and labourers: precisely the people most likely to need help from the church. Food became scarce because there was no one left to harvest it. The number of people left destitute increased dramatically. The Black Death being especially virulent among children, monastery schools closed.
Down at the local level where priests, nuns, and monks were expected to minister to the poor and sick (and administer last rites), the toll was enormous. Some simply fled; the rest were overwhelmed, and the size of the clergy in Europe fell at least a dramatically as that of the general population.
Some historians feel that the dramatic drop in clergy planted the seeds for the Reformation, simply by leaving too few to properly minister.
There is no evidence that any large number of people lost their faith: some did, panicked that nothing seemed to stop the plague, but believers would be very aware of the realities of war and pestilence, and knew that Christian doctrine didn't, and doesn't, promise a miraculous recovery on this earth. ( There never has been any doctrine that the "the church was infallible": in fact, the notion of papal infallibility, something very different, and very rarely applied, only developed around 1870.)
To quote one history site (linked below) "The results of the Church's role in the Black Plague are unclear. Church documents note an increase in gifts to religious institutions, but a shrinking number of churches. Historians point out a general decline in moral standards, but at the same time, a flowering of personal piety and a revival of individual spiritual fervour."
