Should 企 be pronunciated as ㄑㄧˋ or ㄑㄧˇ?
Yes and no.

If you check out Liang'an you'll see it mentions:


  qì


and


  [陸] qǐ


So basically what that tells us is: Mainland (i.e.: PRC) is qǐ, while Taiwan is qì.

It's a regional -or- topolectical thing, rather than a character with different pronunciations at different times.