last updated: 06/12/18
I'll be honest: I actually wrote this on Reddit, but it's something I'd like to be able to reference easily in the future because this topic comes up relatively frequently when someone mentions Vivaldi, so I'm putting it on my website as well.
You can't build Vivaldi yourself from source because Vivaldi isn't open source. Part of it is, specifically, the changes the Vivaldi devs make to Chromium are open source under a BSD license. But the frontend, that's to say, the ui, is not open source. But you can still read and edit the ui's source, you just can't repackage the changes you make and distribute it. That's not to say you can't share your modifications, but I'll talk more about that in a bit.
/opt/vivaldi-snapshot/resources/vivaldi/ depending on which release cycle you're using.
Here are some popular examples of existing user mods:
Vivaldi Hooks implements some oft-requested features yet to be addressed by the devs.
Automatically toggles Vivaldi's interface when the user's cursor isn't near the top or bottom of the window.
Revert the panel to its look during Vivaldi's alpha stages, because some people preferred that.
Attempts to integrate Vivaldi with Arc, a popular GTK theme.
You can even contribute to the project, as in this post where a user made changes to Vivaldi that improved performance and submitted them to Vivaldi staff, who then merged it. I don't know of any other specific examples, but according to this Vivaldi dev on Reddit, other users have submitted changes to him that he ended up merging.
So, again, no, Vivaldi is not open source, but only in a strictly legal sense. You can easily access and read Vivaldi's source, so it's easy to confirm that Vivaldi Technologies isn't doing anything obviously shady with users' data. Which means, really, that the only reasons someone would care about Vivaldi's license is ideological, or brand loyalty to the company that made their preferred browser, and not out of concern for their privacy or security.
And just one last thought, I would certainly prefer if Vivaldi were open source, but I'm not a free software activist, I'm a user. And as a user, it's more practical for me to use Vivaldi for its features than to settle for an inferior product simply because it is open source.