[Typescript Conversion: Level 18]: Convert app/scripts/controllers/permissions/background-api.js to typescript

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## **Description**

Convert `app/scripts/controllers/permissions/background-api.js` from JavaScript to TypeScript as part of the TS migration initiative.

## **Scenario (for user stories only)**

_(Not applicable)_

## **Design (for user stories only)**

_(Not applicable)_

## **Technical Details (for technical tasks only)**

- Convert this file to TypeScript.
- Migration Level: **18**
- Dependencies:
- None
- Dependents:
- app/scripts/controllers/permissions/background-api.test.js
- app/scripts/controllers/permissions/index.js

## **[Threat Modeling Framework](https://github.com/adamshostack/4QuestionFrame) (for technical tasks only)**

- What are we working on? Convert file to TypeScript.
- What can go wrong? Type errors, integration mismatches, missing types.
- What are we going to do about it? Write types carefully and test thoroughly.
- Did we do a good job? _(To be evaluated)_

## **Acceptance Criteria**

- File is fully converted to TypeScript.
- Tests pass.
- File adheres to linting rules.
- Functionality remains unchanged.

## **Stakeholder Review (for MetaMask employees only)**

- [X] Engineering
- [ ] Design
- [ ] Product
- [ ] QA
- [ ] Security
- [ ] Legal
- [ ] Marketing
- [ ] Management
- [ ] Other

## **References**

- Parent issue: #31460

Start with app/scripts/controllers/permissions/background-api.js and review its dependent files, app/scripts/controllers/permissions/background-api.test.js and app/scripts/controllers/permissions/index.js. Convert the file while preserving behavior, then run the related tests and linting; done means the TypeScript file is complete, tests pass, and functionality is unchanged.

Written by the indexing model from the issue text.