Some of the most powerful integration surfaces with Front are Application Requests, Application Objects, and Plugins. 
A framework I find helpful in thinking about when to use these is;

• App Request: Trigger something external from Front (e.g. update a record in your CRM)
• App Objects: Present a view of something external in Front (e.g. show live order statuses on conversations where an order ID is mentioned)
• Plugins: Allow your users to view external data in the plugin sidebar (e.g. based on the customer’s email, show their account details and recent orders. Allow searching your own DB right from a mini-UI in Front)

App Objects are a great place to start. 

As an example; You can have them applied to conversations where someone mentions a shipment by ID (such as “SHP-123456). Have your shipment tracking system trigger calls to the Front Links API on any update to that shipment; this can allow you to immediately update that App Object; and ensure Front users are always viewing real-time information about that shipment on any conversations linked with the App Object. 

Front’s APIs are fairly wide, so there’s a ton of integration approaches you can take. Some good questions to ask your team are;

• What would be useful to see from <our platform> in Front? (Where do I find myself tab switching between Front and <our platform>
• When I have to update a record in <our platform> - could this be automated via a Front Rule or Macro and an App Request (turning it into something we can manage with a couple of clicks from the email in Front)

I hope this gets you started with some inspiration!