Your DOS pc will need the following:

Supported NIC
DOS Packet driver
Microsoft Network Suppliment Diskette set for DOS

the network suppliment for dos has dos NDIS packet drivers baked in, but does not know about all possible cards. It has the option to load an unlisted driver during installation. If you need one for your card, you will have to hunt for it.

this is the basic process:

http://www.kompx.com/en/network-setup-in-dos- … work-client.htm

The ultimate goal, is to get the dos system configured to either grab an IP Address from DHCP, or have one manually assigned in your subnet, then use the NET command to mount file shares or printers.

Bear in mind, that the packet driver, and the protocol stack, consume A LOT of memory.

Try find one of the old ultimate dos network boot disks.
See if it works.
If it does, copy the relevant bits over