Ah, that's a good point. I'd think it wouldn't as well, as it requires IE7.
Quite the pickle, as everything else concerning Demigod appears to work fine, but you can't update it. 
I did however find this:
"For now, at least, SDCentral provides the same functionality, works perfectly under wine, and is still available from this link: sdc.stardock.com/sdcentral_setup.exe"
from http://appdb.winehq.org/objectManager.php?sClass=version&iId=13986
I don't know what SDCentral is or does, but this comment seems to imply that it can provide the same functionality as Impulse. Maybe it allows updates.