This can become quite costly in terms of performance if you receive many requests. Generally these days I would advocate using the App pool user for the site in question. It exposes too much information about your server that hackers could use to attack you.
I had to do some extra work because the default web. Indeed, in many cases you'll find that PHP is running as this user and thus it would make sense that it should have access to folders it wants to write.
So perioically I find myself running into this issue on a Wordpres IIS installation, whereby I'm unable to update plugins or do other things that involve the system putting files into the wp-content folder.
Or you prefer the reliability of a solution provided by the big company that Microsoft is. Keep in mind that putting a phpinfo.
While this is always the first place to look, it's not always the solution for Wordpress plugin update issues. Procmon is a lovely piece of software that gives an incredible amount of realtime information and will allow you to get an idea of what's happening and to which files, when Wordpress is attempting an auto update or plugin update.