Well, I have not read much on Muromachi period prostitution, but I have read quite a bit on Tokugawa era prostitution (17th century). For Edo-period prostitution, Amy Stanley's Selling Women: Prostitution, Markets, and the Household in Early Modern Japan is a great place to start and is a pretty interesting read.

At least in Tokugawa Japan, a woman could end up there for any number of reasons (I imagine that they are not too far off or too different than in the Muromachi period). For some, it was a viable way to make a living and support one's self, so there are cases of women choosing to work in brothels more "willingly.' Daughters were also given to brothels (or "sold"), but sometimes this action was seen in a positive light. Because having to sell your daughter was such a viable way to make money for families that were down own their luck, the Tokugawa government attempted to "reinvent" so to speak, the image of these women such that they were considered virtuous daughters, because they were helping to support their families. But, as one can imagine, this is a rather difficult endeavor and many families were nevertheless embarrassed by their need to sell their daughter and were shamed (to an extent) by others [edited because newmuze made me realize that I needed to clarify].

With regard to being "forced" (in the more violent sense), there are many cases of women being kidnapped and sold to brothels. In Tokugawa Japan, there were laws against this and there were efforts to procure these stolen women.

What I always thought was particularly interesting (and I believe it is mentioned in the Stanley text) is that women could be sent to brothels as punishment, usually punishment for sexual promiscuity. For instance, a father sent his daughter to work in a brothel because she attempted to elope with her lover. Unfortunately, I asked my Professor if there was more literature about "sex as punishment" for sexual promiscuity, but apparently not (at least in the English-language).

But any case, these are some of a few reasons why a woman might end up in a brothel in Tokugawa Japan. One crucial difference that is worth noting is that in the Tokugawa period, the government attempted to restrict prostitution to certain areas of town (largely unsuccessful) and there were generally more laws and restrictions on prostitution. Because of these restrictions, women in brothels had work contracts that could be invalidated, for instance, if it turns out that the person who sold her was not family, though people still slipped through the cracks. But despite this difference, I still think that a poor economic background (a common reason in the history of prostitution) would be the predominant reason that a woman ended up in a brothel.