Edit: Shared not because it's particularly relevant for the legal aspects, just for some relative idea of what it would cost if you did pay for a license from the party that's unhappy.
https://fonts.adobe.com/fonts/proxima-nova
> Licensing Information
> The full Adobe Fonts library is cleared for both personal and commercial use.
> As with everything from Adobe Fonts, you can use these fonts for: ... Website Publishing: Create a Web Project to add any font from our service to your website
Either Adobe has a license, and the foundry is fucking up, or Adobe doesn't have a license, and Adobe has fucked up. In neither case should this be OP's problem.
Agencies used to get in trouble all the time back in the day for embedding font files in client websites that they had only licensed for offline use in Illustration/Photoshop/etc.
It doesn't say "...add any font from our service to websites for your clients", though.
Meanwhile, the exact same font is $30 for a perpetual desktop license. That allows you to create a logo in Photoshop and use that on your website - or use it for a printed document with tens of millions of copies. Or you can get an App license, which allows you to use it in a single app for $500 / year - regardless of the number of installations.
Web fonts are just a complete ripoff, it seems. Considering font faces can't even be copyrighted in the US, you'd be better off hiring someone to make a copy by tracing it!