Does Fox-Fire choose a target?

Yes, Fox-Fire targets.

According to Rule 355.10, a game object mentioned in the text of a spell is a target unless it meets specific exceptions. Fox-Fire instructs you to "Kill any number of units at a battlefield," and per Rule 355.11 and the associated FAQs, these units are chosen and locked in as targets when the spell is put on the chain.

Key points regarding Fox-Fire's targeting:

• Targeting Requirements: You must choose a group of units at a single battlefield whose total Might is 4 or less at the time you cast the spell.
• Lock-in: These targets are locked in when the spell is put on the chain.
• Resolution: If the Might of your chosen targets changes before resolution (e.g., via a reaction), you must choose a subset of the original targets that still fulfills the "total Might 4 or less" requirement upon resolution. You cannot add new targets that were not part of the original group.
• Battlefield Restriction: Because it targets units at a battlefield, all chosen units must be located at the same battlefield. If you play Fox-Fire from hidden, it can only target units at the battlefield where it was hidden.