Resolve "Add eslint-vue-plugin to our stack"

What does this MR do?

1. Adds https://github.com/vuejs/eslint-plugin-vue
2. Fixes errors
3. Updates docs

Does this MR meet the acceptance criteria?

• Changelog entry added, if necessary
• Documentation created/updated
• API support added
• Tests added for this feature/bug
• Review
  • Has been reviewed by UX
  • Has been reviewed by Frontend
  • Has been reviewed by Backend
  • Has been reviewed by Database
• Conform by the merge request performance guides
• Conform by the style guides
• Squashed related commits together
• Internationalization required/considered

What are the relevant issue numbers?

Closes #34312 (closed)