fix: mark stale action couldn't read contents (#41533)

## Description

I'm not sure how this action ever worked, but it's failing to checkout
the codebase because it doesn't have permissions to do so.

> [!TIP]  
> _Add a TL;DR when the description is longer than 500 words or
extremely technical (helps the content, marketing, and DevRel team)._
>
> _Please also include relevant motivation and context. List any
dependencies that are required for this change. Add links to Notion,
Figma or any other documents that might be relevant to the PR._


Fixes #`Issue Number`  
_or_  
Fixes `Issue URL`
> [!WARNING]  
> _If no issue exists, please create an issue first, and check with the
maintainers if the issue is valid._

## Automation

/ok-to-test tags=""

### :mag: Cypress test results
<!-- This is an auto-generated comment: Cypress test results  -->
> [!CAUTION]  
> If you modify the content in this section, you are likely to disrupt
the CI result for your PR.

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## Communication
Should the DevRel and Marketing teams inform users about this change?
- [ ] Yes
- [x] No


<!-- This is an auto-generated comment: release notes by coderabbit.ai
-->

## Summary by CodeRabbit

* **Chores**
* Updated GitHub Actions workflow permissions to support continuous
integration and automation processes.

<sub>✏️ Tip: You can customize this high-level summary in your review
settings.</sub>

<!-- end of auto-generated comment: release notes by coderabbit.ai -->