A host can access the internet, but can't ping any host on its local network. When using tracert it looks like packets meant for local networks are being sent to the ISP. How to solve?

I'm currently setting up an email server for my job, and got it so I can send emails. But only when I'm using my data, and presumably other external networks. The server can be pinged, but it can't ping any device except the default gateway. When using tracert the packet will get to the default gateway just fine, but will then go on to the ISP instead of the intended target, where it appears to be discarded because it is a 10.x.x.x destination address.

I believe the router's routing table is incorrectly configured. My boss doesn't want me messing with router settings because the phone systems are set up there and the phone system cannot go down, so I would like some type of confirmation this is likely the case before we get the ISP out here. Any help is appreciated.