I reckon that ふい is a kango word (不意), but I usually see it written in hiragana, even in print. Neither of the two characters are hard, and if people are typing it shouldn't matter that ふい is easier to write, so I wonder if there are any other reason why ふい is usually writtein in hiragana.
There are two different ふい listed in dictionaries:
- 不意 (jisho.org, デジタル大辞泉): sudden; abrupt; unexpected; unforeseen
- ふい (jisho.org, デジタル大辞泉): coming to nothing; ending without result; total waste
The second one has nothing to do with 不意, and has to be written in hiragana. That is, you can say 不意に思い出す but not チャンスを不意にする nor チャンスが不意になる.
According to this, ふい meaning total waste may be etymologically related to 吹く.
In addition, ふいに meaning unexpectedly is often written in kana these days, and I think that's because this ふいに is becoming a lexicalized adverb, which should be generally written in kana. (We still can say 不意の～, but I have never heard ～は不意だ.) 次第に and 遂に are other examples of adverbs which are commonly written in hiragana.