Your note indicates that these LANs are located at the same location. If so this could just be a routing problem. With some NAT if both networks share the same subnet. But you'll likely need routers that are more capable than average consumer equipment or a layer3 managed switch that supports static routes if you don't need NAT.
Basically just need a static route in each router to the other routers subnet. This can be done over cat5e type cable if they two locations are close enough. For longer distances this would require either fiber or metro ethernet if that's available in your area. This will provide much better performance compared to a VPN solution and leave your Internet connection free to do other things.
Depending on your routers you could also use dynamic routing protocols to provide some WAN redundancy for egress traffic to the Internet at least. For example if LAN 1's Internet connection was down the traffic could be sent to LAN 2's router and use it's Internet connection and vice versa.