I like the approach that the kanji useing website I am useing [0] took to input. They accept the input itself to be in romaji, by transliterate into hiragana as you type.
That is good when the input is known to be Japanese, but in this case the user might be inputting Japanese using the English keyboard, or might be inputting an English word to look it up in Japanese.
[0]https://www.wanikani.com/