I've seen that website before. Very interesting take on post office closings around the US.
Back in the 1950's when post offices were known to close, the Postmaster would accommodate philatelic requests to cancel covers with the "last day of use" postmark such as in this Massachusetts example (taken from Buck-a-Cover's website):
It seems a shame the USPS isn't promoting similar philatelic offerings for the current round of USPS closings, although with the sporadic closings and the community protests going on to "fight" for certain post offices to stay open, the actual date of closing can be very fluid, so maybe that has something to do with it.
I would still find it interesting to acquire some Last Day of Use postmarks. Anyone have ideas as to how to best approach a post office with such a request?