Or was everybody all but covered in so many bug bites that people stopped noticing the itching, similarly to how people could ignore the smell in medieval cities?
Lice, mosquitos, and other dirty itchy things were both endemic and problematic in the medieval world. In a passage in his Etymologies which will get picked up by medieval Latin medical authors, Isidore of Seville (d. 636; patron saint of the Internet) lists nine types of "vermin of the flesh", including ticks, bedbugs, various worms, fleas, and lice. Lice (pediculi) are named that because of their feet (pedes), whence those are called "lousy" (pediculosi) on whom lice "boil up" on the body. Ew, right?
Medieval people thought so. They understood that lice infestations spread among people in close contact and with poor hygiene. In 1296, for example, Bernard de Gordon associated lice particularly with the urban poor and members of monastic orders, groups known for living densely and taking less care to bathe (among other things, not washing regularly by choice was perceived as a demonstration of ascetic spirituality and holiness. The wearing of hairshirts as, yup, another practice of asceticism was likewise common among medieval religious men and women and would have allowed for vermin infestations as well).
People tried means both chemical and mechanical to rid themselves of lice. The narrowest-toothed combs discovered in archaeological digs tend to have remnants of lice trapped between the teeth. This matches surviving woodcuts of women combing men's hair that illustrate the "pediculi" passage in the earliest printed medical texts.
Medical texts record a range of options for dealing with lice. The Trotula, a 12th century anthology of medical and hygenic advice famous for its focus on women and partial female authorship, says that lice in pubic and armpit hair are best dealt with through a mixture of ash and oil. Lice on the head require a mixture of aloe, mercury, and incense.
Mercury in particular was a commonly cited cure by the late Middle Ages, and in fact, is one of the ways we can connect medical texts with medical practice. (NOT always a given in the Middle Ages. See also: one of the prescribed means of contraception in the Islamic medical tradition is for a woman to spit three times into the mouth of a frog.) The body of Ferdinand II of Aragon, who had much more success as the king of Naples, was discovered to have both a massive lice infestation and signs of treatment with mercury. Archaeologists posit that the mercury was indeed applied in an attempt to treat the lice. (Fornanciari et al., 2009)
Medieval people likewise recognized that vermin were pests to animals. Isidore notes that ticks are particularly a problem for dogs, for example. In the Islamic world, the leaves of the herb rue were juiced and then sprinkled over chickens as a mosquito repellent.
One thing becomes clear with the exploration and colonization of the New World, though. Bugs and mosquitoes in the eastern Hemisphere had nothing on their relatives in the Americas. In Cuba:
They found immense annoyance from mosquitos...THe insects came in great numbers and settled on the person where exposed, their bites raising lumps that smarted keenly. A single blow with the hand sufficed to kill so many that the blood would run over the arms and body.
And Canada:
I thought I would be eaten up by the mosquitoes, which are little flies, troublesome in the extreme...They are disagreeable beyond description. No one could work, especially in the open air, during their reign, unless there were smoke near by to drive them away. Some people are compelled to go to bed after coming from the woods, they are so badly stung. I have seen men whose neck, cheeks, and whole face were so swollen that you could not see their eyes.