Resolve "Search merge-requests with API"

What does this MR do?

Officially adds search param to the Merge Requests API

Are there points in the code the reviewer needs to double check?

Why was this MR needed?

Please see #38318 (closed)

Screenshots (if relevant)

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 #38318 (closed)