Is there any way of taking a screenshot of a website in PHP, then saving it to a file?
You can use simple headless browser like PhantomJS to grab the page.
Also you can use PhantomJS with PHP.
Check out this little php script that do this. Take a look here https://github.com/microweber/screen
And here is the API- http://screen.microweber.com/shot.php?url=https://stackoverflow.com/questions/757675/website-screenshots-using-php