LAN B is on 172.13.x.x (this is our robot, our PLC etc)
172.13.x.x is not a private-use network range. It's an address range that belongs to AT&T (US), for their residential and business customers (including some web servers). Don't create future headaches for your clients' IT teams by asking them to route a large chunk of Internet addresses to your PLC network, just for you – and even if the PLC network is completely isolated from the Internet, don't make them wonder "why is our PLC talking to someone in Texas".
The 172.16.x.x ~ 172.31.x.x range is reserved for this use.
I need our PLC (so only one device on our network) to be able to speak to their two PLC's and don't know how to do it.
I don't know what kind of configurability typical industrial routers offer, but generally speaking, a router isn't limited to just LAN-WAN, general-purpose routers can be configured with any number of networks – their operation doesn't rely on Internet, they treat it (or its absence) like a network like any other. So allowing two subnets to talk through a router is something that a router pretty much does by default – that's literally its primary function.
But the issue with regular routing is that both sides need to know a route towards each other. So those two PLCs (or, more likely, the 172.16 network's gateway that the PLCs use) would need to have a static route defined for 172.13 through the new router. That would need to be arranged with the network's admins.
So the simplest option in your case – though not necessarily great – is actually to treat the 172.16 network as a "WAN" in terms of router configuration. More specifically, have NAT enabled on the router so that it hides the 172.13 "LAN" from the "WAN" side – all communications would appear as if coming from the router's own 172.16 "WAN" address, so that no configuration is needed from the existing network admins at all.
Many routers (like your home or office router) work in this mode by default – that is, have a "LAN" + "WAN" style setup by default – and I assume there are industrial ones which do as well, but if not, then NAT is probably available as a checkbox on most of them anyway.
