dont know about a designated storage facility.......
I often park up at the Poseidon Center shopping mall for 3 or 4 days at a time without any problems.
might want to ask at security if it's OK.
it's about halfway between Palermo & the airport.
I wouldn't feel bad about leaving the van at the airport parking.....
it's way out in the boonies & unlikely to come to any harm.
just my 2 cents.....no guarantee implied.