[FEATURE]: Adding a dropdown search suggestion

### Feature and its Use Cases

When a user types in the search input, the app should:
Call GitHub organization search API
Show a dropdown list of matching organizations
Allow users to click a suggestion to add it as a chip
Reduce typing effort and improve discovery

### Additional Context
Something like this->

### Code of Conduct

- [x] I have joined the [Discord server](https://discord.gg/hjUhu33uAn) and will post updates there
- [x] I have searched existing issues to avoid duplicates

Start at the search input and trace the existing organization lookup, then read the GitHub organization search API usage and the chip-creation flow. Done means typing requests matching organizations, suggestions appear in a dropdown, and clicking one adds it as a chip.

Written by the indexing model from the issue text.