If you miss the deadline for reimbursement from flexible benefit
account because of family emergency, is there a mechanism for
extension, and what is it?

They don't accept any excuses for missing that deadline.
You're stuck.

If the charges were made in the correct time period, in my experience
you could talk to the plan administrator to see if they will extend
the reimbursement date for you.  This is usually up to the plan
administrator?s discretion. Many times a plan will give a 6 week grace
period.    The key is to make sure that the actual charges are in the
correct time frame as this part is mandated by federal law.