technology recommendation for LAN Dashboard
If you're comfortable with a LAMP-style stack with PHP, then there's no reason you can't use either Django or Rails.  Both are mature, well documented platforms with active, helpful communities.  
Based on what you've described, there's no reason that you can't use either technology.
I'm about to start a fairly large project for a mid-sized business
with a lot of integration with other systems (POS, accounting,
website, inventory, purchasing, etc.) The purpose of the system is to
try to reduce current data siloing and give employees role-based
access to the specific data entry and reports they need, as well as to
replace some manual and redundant business processes. The system needs
to be cross-platform (Windows/Linux), open source and is primarily for
LAN use.
My experience is mostly PHP/web/app development, but I have developed
a few LAN apps using Java/Servoy (like Filemaker).  I found Servoy to be very rapid and to easily make use of different data providers (DB products), but it's not open source, and any non-standard development is in Java/Swing (which is verbose and takes a lot of time).  
I'm interested in learning Python/Django or Ruby/Rails - but I'm not sure if these are the best solutions for building a mission critical data entry/reporting LAN app.  Is a web client/server really a good choice for this type of application?  
Thanks in advance for any tips/ advice.