It looks like the FastCGI proxy can’t open the unix socket because of a permissions issue. This might be caused by the user your PHP-FPM is configured to listen for is not the same as the user Caddy is running as. In PHP’s www.conf, there should be listen.user and listen.group settings for the socket.
@Whitestrake Thank you, that was it! I ran sudo usermod -a -G www-data caddy to add my caddy user (I have a user specifically for Caddy to keep stuff compartmentalized) to the www-data group and uncommented the line listen.mode = 0660 in /etc/php/7.0/fpm/pool.d/www.conf, and now it works!