Is there a common browser shortcut for jumping to a website’s search bar?

Not in every situation but / (slash) works on a lot of websites like Google pages (Youtube, Gmail, Drive, Photos...), Facebook, Twitter... and brings you to that page's search box

/ - Search

How can I navigate Facebook with keyboard shortcuts?

In Github you can even see the / shortcut on the search box

s or /: Focus the search bar. For more information, see "About searching on GitHub."

Keyboard shortcuts

Jenkins:

/ move cursor to search box

Same to JIRA:

See Jira - Using Keyboard Shortcuts

Infoplus:

Tip: Use the Forward Slash (/) to Access Quick Search

It also works in Google after searching: Shortcut for Google search box

/ is also the shortcut for "quick find" in Firefox.
If you want to do the same in Chrome you'll need SlashSearch

This key probably comes from several Unix command line tools (lile less or vi) where / is the search feature

Since you are using chrome, the closest thing I can think of is using an extension called vimium.

1. You press "f," it shows a letter (or two) in front of every field on the page.
2. You press that letter, and it simulates a click.

This way can achieve the desired outcome of using a shortcut to jump to the search bar, but this shortcut changes from page to page.

example : in this case you press: f followed by d

If you're using Firefox, you could go to about:config and configure accessibility:tabfocus to 1. See here for more details: http://kb.mozillazine.org/Accessibility.tabfocus

This will force the Tab key to give focus to text fields only so most of the time, just once or few presses of the Tab key directs it to the search bar and other text fields.

This userscript adds a slash keypress event listener for google search pages.

You can use it with Tampermonkey in most modern browsers.