JetBrains: Wire up a basic version of the search box component

Created by: philipp-spiess

Part of #34366 (closed)

With a working web view and TypeScript setup, we're now able to import the web React components and render them inside JetBrains!

Test plan

Manually verified the changes:

App preview:

• Web
• Storybook

Check out the client app preview documentation to learn more.