So I read https://developer.valvesoftware.com/wiki/$envmap and followed it to the letter, but any custom cubemaps I make won't show up on my model! Apparently I was able to do this like a year ago but now I can't. That's how I know the issue is with the vtf and not the vmt.
Below are my steps, please tell me if I'm missing any critical step:
1. Import 6 32x32 pngs into VTFedit as an environment map. No mipmap, no level of detail, DX1, DX1.
2. Select Clamp S and Clamp T (doesn't seem to make a difference)
3. Save as new vtf file
4. Reference file in VMT as the $envmap texture

EDIT: It turns out all I had to do was reinstall VTFedit (original was 132, upgraded to 133) and it works!