Will a company firewall prevent me from using the Dropbox feature?

Potentially.  You will want to make sure your company firewall is not blocking links to Dropbox.

You will need to request your IT group verify the VPN is open to Dropbox files. To do this, they will need to allow access to *.dropbox.com (meaning all addresses ending dropbox.com) AND *.dropboxusercontent.com (meaning all addresses ending dropboxusercontent.com).  While they are making changes, be sure to allow access to *.foreflight.com (meaning all addresses ending foreflight.com).

Once that's in place, you should be able to use the ForeFlight Mobile app to access the documents you've put into your Dropbox/Apps/ForeFlight folder.