A name for hat collectors?

There isn't really a specific word for someone who likes/collects hats (i.e. - you won't find one in a dictionary). But people who are interested in hats may well know that...

A hat maker is called a milliner.

...in which case they would doubtless understand this coinage that I found in Google Books...

Sadly, living in sunny, funny LA, l have to wait for bad weather to visit the milliner. lf you're a millinophile like me, don't miss The Millionairess.

Since wealthy people could more easily afford to buy lots of expensive hats, it's at least possible that "The Millionairess" is a deliberate play on words there, but probably it's just a coincidence.

capel is the Latin word for hat, headdress.

For a neologism, capellophile is a possibility, and seems to be in limited use in French slang already.