Fill "in" or "out" a form is OK. However, it is always "fill IN the blanks" is the correct form. English is "weird" in this sense!

Both are used nowadays. However, when I was growing up, only 'fill in' was acceptable in British English. I recall an occasion during my teenage years when I was doing a German exam. I had to translate a text which included the German verb 'ausfüllen'. When you translate the constituent parts of this verb into English, you get 'fill+out'. I wrote 'fill out' in my English translation and failed. My teacher was adamant that what I had written was wrong. "Fill out" is not English, she told me. It's an interesting fact that when you compare British and American English, many features of AmE are closer to Germanic-style English. So it came as no surprise to learn that 'fill out' is used in American English. And, equally unsurprisingly, with the growing globalisation of English and the increasing dominance of American English, 'fill out' is becoming more and more common worldwide. Nowadays the internet tells us to fill in gaps and blanks, but fill out forms. It still rankles every time I see 'fill out'. Not because I feel it's wrong, but because I resent failing that German test all those years ago!

"The London England perspective" is Either fill in or fill out, Although for an older speaker like myself "fill in" is more instinctive. It is what I was bought up with. Now in LONDON "multi cultural London English". Nobody bothers anymore to argue over it. Both are used currently.

Fill in = British Fill out = American