'Ello folks. I recently got all of the parts in necessary to rig a desktop GPU to my T420, but for some reason it isn't quite working. Everything is plugged in and powered correctly, and the drivers for the card (AMD R7 240) are installed, but my external monitor refuses to recognize that it is plugged in.

Hi & WELCOME to this site! Sorry that your question didn't "ring a bell" with anyone.
I'm not familiar with any of this, but hopefully I can contribute some general troubleshooting ideas here (IF it's still not working).

I'm guessing this is basically a 'second linux graphics card', so I googled that & learned of
xrandr! I already was thinking of your posting a couple relevant lines of lsmod/lspci/dmesg
and a relevant bit from the {X,x}org .log / .conf{,ig} ... [I don't know specifically]
to give the experienced helpers "something to go on".
Also maybe, how the eGPU connects to the T420 (and, IF you use MSwindows, does it work there).

Best wishes! p.s. post a solution to help others if you fix it.

corvus,

Welcome to LQ.

Did you manage to find a solution?

If not, try looking at this:

https://www.youtube.com/watch?v=Ka5gBMlo-zo

It may give you some ideas for your Arch installation.

Is the monitor plugged into the card??
Can you draw a diagram of how you have it set up??