What is the correct te-form of 問う? Is it 問って or 問うて or both?
While 「問って」 may seem the logical conjugation, 「問う」 is actually irregular (see the Wikipedia entry for 不規則動詞). According to this article, 「問って」 is "almost never used". It appears therefore that 「問うて」 is correct in modern Japanese.
In case you are wondering why, the author of the latter article hypothesizes that this irregular conjugation makes the dictionary form of this verb more obvious when using its te-form in speech (as well as others, such as 「乞う」).
One contributing factor is that the pronunciation of the dictionary form (問う) can be thought of as one long vowel syllable (トー), so conjugating it as 「問って」 (トッテ) would result in the modification of that identifying first syllable.
Of course, this reason alone would not normally be sufficient justification for this irregularity. The author adds that due to frequent substitution of 「問う」 with 「頼む」 in modern Japanese, 「問う」 is now less common and thus requires extra disambiguation when used in the te-form.
問う has a hidden sound "w" at the end of the verb stem, which does not arise at the surface in nonpast forms due to a phonological rule that deletes "w" in front of vowels other than "a". 買う is another such verb:
tow-u → tou
kaw-u → kau
So for the te-form, you would expect gemination, which happens with other verbs ending in "w". However, as written in the wikipedia link that con5013d cites, This verb is irregular, and the form
toute (← towute) is used.
kaw-te → katte
* tow-te → totte