I want to start an upgrade on my Archer Queen during a Clan War, but need her to attack in the war. Will the Archer Queen function during a clan war while being upgraded?
No, Heroes are not usable in any circumstance when they are being upgraded, and this includes the Archer Queen. If you want to use your Archer Queen during a war while she is being upgraded, you will have to speed up the upgrade time either by using a Magic Item (either Book of Everything or Book of Heroes or one or more Builder Potions [lasts for 1 hour and speeds up upgrade time by 10, so you effectively save 9 hours], or a Hammer of Heroes) or use Gems to instant complete the upgrade.
Note that heroes are not considered buildings however and as such, the Book of Building and Hammer of Building will not work on a Hero's upgrade.
As a side note, the current cost in gems for hero upgrade I believe maxes out at 999 gems for 7 days' upgrade (I can't find a reliable source for that, but my own experience), so if you are planning to use gems, ensure you will have enough.