Vintage Mac OS7 to router connection problem

[#1]

[#2]

And your problem is.....?
'Gis a clue.

[#3]

Does OS 7 even support TCP/IP?

[#4]

Yes, I believe it should do, though OS 7 was a *long* time ago - your first port of call should be the Control Panels, probably one called "TCP/IP". Have a read of this thread for some tips.
You might be better upgrading to at least 7.5.3 which had better networking capabilities if I remember correctly. Even better, OS 8 or 9 if your Mac is capable of running either of those OSes. Think I have some old System 7.5 discs lying around, as well as OS 8.
Can you tell us what model of Mac it is, please? And how much RAM it has (go under the Apple menu and select "About this Macintosh").
Not all Macs of this vintage had network cards installed, and even if they do it might not be Ethernet - at that time it was LocalTalk wasn't it - like I say, my memory's a bit hazy now.