Try Chicken Little—quite popular in this forum…

Christoffer, I'm not aware of a similar webfont. Stag Sans has some very unique characteristics. If you remove those, you might arrive at something that resembles a Franklin Gothic or Trade Gothic (maybe). But, I'm not sure you'll find anything very close. I would contact Commercial Type and see if you can license it as a webfont.

Chicken Little was not even a close match for Stag Sans. I do think if you are in a jam Calibri will work unless you have the budget to actually purchase Stag Sans.

To me, it seems like using something like Trade Gothic for the light weights, and Antique Olive for the bold weights, would be vaguely similar.

I know the x height isn't as big and it runs a bit condensed comparatively, but have you tried Lucida Sans?