Yes you can.
Looks like your Procfile is malformed (assuming you copy/pasted correctly). It should read:
web: vendor/bin/heroku-php-nginx -i custom_php.ini -C nginx_app.conf
It might be a good idea to copy the default php.ini (https://github.com/heroku/heroku-buildpack-php/blob/master/conf/php/php.ini) to custom_php.ini and add your customizations.
You can also quickly and easily re-define the document root like this (to folder public/ inside the app):
web: vendor/bin/heroku-php-nginx -i custom_php.ini -C nginx_app.conf public/
By the way, if you have PHP 5.5.11+ and Nginx installed locally, try composer require --dev "heroku/heroku-buildpack-php *" and then foreman start. That should boot a server on localhost, port 5000, using your Procfile and with all the same settings as on Heroku.