In VaultWiki, the button is designed to be inside the search box, not next to it. The issue has nothing to do with sidebar or responsiveness or anything like that, but that the class for the button makes the button larger in height than the height of the class for the search box, resulting in the button overflowing outside the search box it is inside.
There are only two ways to fix this: increase the height of the search box, or decrease the height of the button. Even if the button was next to the search box, rather than inside it, it would still look awkward if they were not the same height.
I think that reducing button size is the best solution because:
- Increasing the height of the search input would be inconsistent with every other input field on the page
- The button does not have the same level of importance as other callToAction buttons or even other form controls on this tab (for example, Save Changes is actually smaller in size in your style), so it could stand to be muted in some way.