[3.11] Docs search: Replace jQuery with vanilla JavaScript (GH-106743) (#106803)

Docs search: Replace jQuery with vanilla JavaScript (GH-106743)

* Replace jQuery with vanilla JavaScript
* Switch 'var' to 'const' or 'let'
(cherry picked from commit c02ee4503151105dc892018ebc7f633e7f3f62f8)

Co-authored-by: Hugo van Kemenade <hugovk@users.noreply.github.com>