So I've just now successfully been able to download, install and play Zzap! on the Itch launcher on two different computers, which makes me wonder if it's something to do with the security settings on your computer preventing installation of the game. This could be anti-virus software, firewall, insufficient user permissions... I'd suggest trying logging in to your computer as admin, launching Itch as admin, checking your firewall settings and allowing Itch as an exception to the firewall, and disabling any additional anti-virus software. I hope you're able to find a solution!
Thanks!