Hello all. I want to make a program in C++ that will retrieve router tables from various portions of a LAN. Is there any way to get this data via a packet?

You would have to advertize yourself as a router information exchange point. Look up what protocol your routers are using (RIP, BGP, whatever) and implement that protocol (or find a library that does) to get that type of information.

It may also be that various devices from various manufacturers expose this information using some management protocol such as SNMP or vendor-private protocols. You have to study the documentation for the specific devices you are interested in to get more detail.