How about Vinter? Manually hinted, but it is so thin you should really only use it for big sizes.

(Edit: My memory of the lightest Domaine weight didn’t match reality. Vinter is too thin. I might have something up my sleeve though, if you contact me directly)

Domaine Sans is such a specific face. However, one of the following might work for you:

Maple:
http://processtypefoundry.com/fonts/maple/

Perry Grotesque:
https://ourtype.com/#/try/pro-fonts/parry-grotesque/

Supria Sans:
http://www.myfonts.com/fonts/hvdfonts/supria-sans/webfont_preview.html

Bureau Grot:
http://www.fontbureau.com/fonts/BureauGrot/

It says coming soon on Klim Type's website and I have been crossing my fingers, but the deadline is closing in and I will have to find an alternative.

Have you asked the foundry about it? Just because the webfont licensing is not yet available as online store functionality doesn’t necessarily mean that the foundry isn’t willing or able to provide webfonts on request.

Thank you for some really nice suggestions. Frode, I've actually already looked at vinter. It's a beautiful typeface. (I'll send you an e-mail) And Ralf, your suggestion is good but I've already been in contact with the foundry and they couldn't give me any estimated time of arrival.

I'm currently testing out Beausite Grand which has some of the characteristics of Domaine Sans although less of that crazy flair. But please, if anyone has any suggestions to add, do let me know.

Perhaps Bodoni Sans Display: http://www.myfonts.com/fonts/greyscale-type/bodoni-sans-display/