Can I install an additional network card on that specific PC, attach a Ethernet cable from it to a broadband router, and thus have Internet access on that machine?
Yes, this would work.
The new NIC would be used as default, reaching any and all destinations,
except those where you explicitly set a route to use the other network card.
Depending on your OS this might be the simple default, or you may need to set a few routes. For those please see this post on subnetting (and routes).
If you connect things like this
and use different network ranges then you can tell the windows PC that all traffic for the 'green network' needs to go out via the green wire and everything else via the red wire.
To be able to reach all locations on the internet the 'green network' should not use any network ranges which are already in use on the internet. If you have your own public IP range then you can use that. If not select a range from RFC1918
(Briefly, that is from 10.0.0.0/8, 192.168.0.0/16 or 172.16.0.0/16)
Personally I am a fan of the 172.16.x.x. addresses since most people seem to avoid then. Thus avoiding potential future conflicts is several networks get merged.
Assuming this is the case and you selected 172.16.67.x/24 for the factory PCs and 192.168.74.x/24 for the windows PCs second NIC then:
All factory PCs check if the network is for themselves, if so loopback.
If not and the destination is the local LAN, send via the network card.
(No default route needed)
The windows PC checks if the network is for themselves, if so loopback.
If not and it is for the 171.16.67.x/24 range, send via the green wire.
If not and it is for the 192.168.74.x range, send via the red wire.
In all other cases send via the red wire (default route).