You need to work on this dialog.
I think it has scared off many customers willing to buy a game but not willing to send their private information to who knows where.

What this dialog is missing:

+ Who gets to see this information? Itch or the person I'm buying from? The credit card processor? All three parties?
+ Where is this information stored? On itch's servers? On the credit card processor's servers?
+ Can I see this information in my account? Meaning: If someone hacks my account, can he then see where I live?
+ How long is this information stored? Forever?
+ Is the information stored encrypted? If someone hacks your database will everything become available to him?
+ Why do I need to enter this information? Linking to ec.europa.eu is not acceptable, you can't expect every customer to study some official law document when it's easier to just not buy the game. At least provide a short explanation why this is needed on itch while you don't get this on other platforms where you can buy games.

The link you have appears to be outdated btw.

http://ec.europa.eu/taxation_customs/taxation/vat/how_vat_works/telecom/index_en...
redirects to
https://ec.europa.eu/taxation_customs/business/vat/telecommunications-broadcasti...

Also please don't just explain these points in the thread, fix the dialog. Make sure the information is there and it's easy and quick to read.

The most annoying part of this is that there is no option to save the delivery address, so I have to fill out the whole form every single time I make a purchase. The lack of a shopping cart makes this even worse.

"You need to work on this dialog.
I think it has scared off many customers willing to buy a game but not willing to send their private information to who knows where."
Yeah, this! Nobody thinks twice about Steam having your personal information but I bet getting flashbanged with a WHERE DO YOU LIVE?! jumpscare turns away many potential customers.

I was in the process of purchasing assets when this box popped up. How should I fill it out to access PayPal normally

I was scared off, and the fact that this has been the same now for 4+ years seems to imply that Itch.io couldn't care less. Please listen to Blodsmak first post. 

For everyone interested, there is some info here:

• 2014 post about VAT MOSS 
• 2015 post about the new payout system
• itch.io creator docs about VAT

TL;DR:

• When selling to the EU, the seller needs to pay VAT in the buyer's country
• MOSS is a system where sellers can report and pay their taxes for all EU countries at once, instead of each one individually
• For MOSS, the seller needs to provide two pieces of evidence showing where the buyer is located
• If the seller receives a direct payment (i.e. itch.io is *not* collecting the payment first), itch.io instead collects the buyer's IP address and billing address as pieces of evidence. The seller can then download that data in bulk to pay their VAT with MOSS

That doesn't answer all of OP's questions, but it helped me understand enough about what was going on. At the very least, linking to those existing creator docs in addition to the EU website would be a lot less puzzling than only a reference to ten year old regulations with a broken link.