🚀 badges: Migrate to Backstage UI

### Workspace

badges

### 🔖 Feature description

The plugin can be migrated to BUI.

The plugin code is here: https://github.com/backstage/community-plugins/tree/main/workspaces/badges/plugins/badges

The list of available Backstage UI components is here: https://ui.backstage.io/

The process would be to just switch out the Material UI (MUI) components with their Backstage UI (BUI) versions and visually check to make sure they match. Additionally the Table component from `@backstage/core-components` can be changes to use the Table from Backstage UI (BUI).

### 🎤 Context

~

### ✌️ Possible Implementation

~

### 👀 Have you spent some time to check if this feature request has been raised before?

- [x] I checked and didn't find similar issue

### 🏢 Have you read the Code of Conduct?

- [x] I have read the [Code of Conduct](https://github.com/backstage/community-plugins/blob/main/CODE_OF_CONDUCT.md)

### Are you willing to submit PR?

No, but I'm happy to collaborate on a PR with someone else

Start in workspaces/badges/plugins/badges and compare its Material UI components with the available Backstage UI components at ui.backstage.io. Review the existing pull request before starting, then visually check migrated components and verify that the Table component uses Backstage UI rather than @backstage/core-components.

Written by the indexing model from the issue text.