"Please write it on the blackboard."
You are correct that 在 ordinary behaves like other coverbs (or "prepositions," if you prefer that term), with the entire coverbial phrase always appearing in front of the main verb. There are a handful of verbs for which 在 can occur after the verb. Ones that immediately come to mind are 住在 "to live at," 放在 "set at/on," and 生在 "be born at." This post-positional pattern also occurs with the coverb 给"give." (送给, 带给, etc.)
在 is mostly optional but there are certain instances in speech where it's needed for clarity. Without 在, this sentence is more often recognized as ”please write (the word) "黑板上“" (maybe on a piece of paper or something). It's not need in "黑板上写什么？“ sentence because there is no other way to understand the sentence other than "what's written on the blackboard?" In this case the 在 is optional so it can also be written as "在黑板上写什么？"