Firefox makes the majority of that income via their Google partnership today. Recently, the majority of it was coming from Yahoo.
The browser's market share will ultimately determine the market value of the search partnership, so Vivaldi will likely have a much lower income from it than Firefox, but it's in no way limited by whether they partner with Google or with any other search provider. For a smaller more efficient team, not maintaining their own browser engine, it could easily compare in sustainability to Firefox's income stream.
There were a couple years recently Firefox was primarily sponsored by either Yahoo or Bing (can't remember which) and that deal supported the entire Mozilla foundation same as the Google deal as well as Firefox (which also makes their own browser back end) not just a browser front-end dev team.
So there are definitely other decent sources of income besides Google for these deals and since it's a significantly more focused organization it doesn't seem implausible to me.
These deals are based on market share, user activity and usage, and so on. No search engine is paying Vivaldi as much money as Yahoo or Google did to Firefox. It would be laughable to expect that kind of sponsor money for a niche browser that also reduces ad tracking. That’s how these things work.
It'd be laughable to expect Vivaldi needs the exact same 600 million dollar search deal that floats the entire Mozilla foundation to fund a custom Chromium UI.
As I said, there are definitely other decent sources of income besides Google for these deals and since it's a significantly more focused organization it doesn't seem implausible to me.
Firefox is funded the same way, and they seem to have plenty of income...