In the sentence この人たちは何歳にみえますか, I originally thought to use the particle を since it marks the thing getting "verbed" (and it seems like 何歳 is getting verbed), but に seemed possible as well, and the website bunpo-check marked を as wrong in this context. I would like to know which particle to use and why. What definition captures what the particle is doing? Is it marking the direct object, the indirect object, or something else? What particle does 見える require and why?
何歳を見えますか is ungrammatical. You must say 何歳に見えますか.
This is because 見える must take an adverbial form of words. Here are possible patterns:
- na-adjective: 簡単に見える looks easy
- no-adjective: 緑色に見える looks green
- i-adjective: おいしく見える looks delicious
- noun: 学生のように見える looks like a student
- verb: 違って見える / 違うように見える looks different
See this external article, too.
I'm not sure what you mean by "verbed", but N歳 works not as a verb but as a no-adjective (you can say 18歳の男 for example).