We know precious little about what peasants thought about most everything in the Middle Ages. For the most part, we don't have sources that describe the beliefs of non-literate people. Likewise, the Middle Ages is not a homogeneous period, temporally or spatially, so we shouldn't suppose that even your average peasant would have quite the same general knowledge in different times or places, let alone the range of beliefs within the peasantry.

In all likelihood, must as for us today, most peasants probably didn't think much about the shape of the earth at all. It is likewise likely that among those who did, some probably did imagine that the earth might be flat. Indeed, this would hardly be the strangest thing, given some of the bizarre cosmologies that emerge in inquisition records. (The most famous, perhaps, is that of Domenico Scandella as described in Ginzburg's The Cheese and the Worms. Although, these records, and Ginzburg, need to be treated with immense care, since inquisition depositions are by no means a direct record of peasant beliefs.)

That said, by the later Middle Ages, what little evidence we have seems to indicate that knowledge of the earth's shape was accessible to the laity and, by all indications, reasonably widespread. We have good reason to suppose that it was accessible to parish priests, even if they weren't really latin literate, in a range of widely disseminated, vernacular encyclopedias. These sorts of encyclopedias will have provided material both for inclusion in sermons and for general discussion with the laity, and all other things being equally, we can reasonably suppose that it will have percolated down to a wide range of people this way.

By the thirteenth century, we begin to have records of vernacular sermons, which are probably the closest thing to mass media in the Middle Ages. We find cosmological details, like the shape the earth, discussed fairly offhandedly in these contexts. For example, within a broader discussion of the structure of the cosmos, Berthold of Regensberg (d. 1272) includes the comment that:

the earth is created quite like a ball

diu erde ist rehte geschaffen alse ein bal

(trans. Hall in Simek, Heaven and Earth in the Middle Ages, 30)

Likewise, one of the jobs of the mendicant orders was the education of the laity, and while this was primarily about religious matters, we know that they also disseminated a range of 'scientific' knowledge of the sort taught in the universities. For example, in the 14th century, Conrad of Halberstadt wrote a little handbook of interesting scientific facts that could be used to engage an interested lay person in conversation. Notable in this context is the fact that the shape of the earth is simply not included, although the shape of the heavens is. This seems to suggest that he thought it sufficiently well known that it would not be a point of interest to the laity at large.

So while we can't make definitive comments about what everyone believed across the Middle Ages, in the places where we do have evidence, it seems more likely than not that this was common knowledge even among those who were illiterate and never formally educated.

(I've discussed this before here, with some more specific discussion of the evidence.)