From my understanding, 伺う is the humble version of 会う, which means to pay someone / some place a visit, and requires a recipient. However, on this website, the humble version of 会う is listed as お目にかかる & お目もじする instead of 伺う.
If I look at the form of each compound verb, the meaning is rather obvious for 御目に掛かる. I believe it literally means, "I place my (humble) presence to your eyes," which makes sense. On the other hand though, 御目文字 is plain puzzling to me. 三省堂's definition says:
If I'm not mistaken, 近世 means *"early modern." By this definition, I assume that it was originally feminine language, but not anymore, is it? What about its kanji form, 御目文字(する), which would be ご-eye-word, I daresay. How does this kanji combination relate to 御目に掛かる? *The English words "modern" and "contemporary" have different usages in different fields of study. Sometimes they overlap, sometimes they don't.
Lastly, how are お目にかかる, お目もじする, and 伺う used differently? Do people write them in full-on kanji, or otherwise?