Is there a way to make editor tabs stack in VS Code?

As of July 14th, 2020 there is no built-in way to make the tabs span multiple rows.

There is a discussion within the VS Code github repository about adding this feature here: https://github.com/microsoft/vscode/issues/70413

On the latest release of VS Code v1.53 this feature has been added

You can add "workbench.editor.wrapTabs": true to your json config file

Or search for Workbech > Editor: Wrap Tabs in Settings UI.

In adition to that you can enable shrink on tabs so they use less space (only the width of the characters) with "workbench.editor.tabSizing": "shrink"