Auto-update Sample App Website (toolkit.dev)

### Describe the bug

(Thought we had an issue for this, but can't find it.)

We should setup a separate workflow which knows how to push to the toolkit.dev repository to update the website with the latest code from main.

We probably just want to be able to trigger this manually vs. on every commit to main (or that we can chron job to happen once a week or something).

### Steps to reproduce

```text
N/A
```

### Expected behavior

Website is kept up-to-date with repo.

### Screenshots

_No response_

### Code Platform

- [ ] UWP
- [ ] WinAppSDK / WinUI 3
- [ ] Web Assembly (WASM)
- [ ] Android
- [ ] iOS
- [ ] MacOS
- [ ] Linux / GTK

### Windows Build Number

- [ ] Windows 10 1809 (Build 17763)
- [ ] Windows 10 1903 (Build 18362)
- [ ] Windows 10 1909 (Build 18363)
- [ ] Windows 10 2004 (Build 19041)
- [ ] Windows 10 20H2 (Build 19042)
- [ ] Windows 10 21H1 (Build 19043)
- [ ] Windows 11 21H2 (Build 22000)
- [ ] Other (specify)

### Other Windows Build number

_No response_

### App minimum and target SDK version

- [ ] Windows 10, version 1809 (Build 17763)
- [ ] Windows 10, version 1903 (Build 18362)
- [ ] Windows 10, version 1909 (Build 18363)
- [ ] Windows 10, version 2004 (Build 19041)
- [ ] Other (specify)

### Other SDK version

_No response_

### Visual Studio Version

_No response_

### Visual Studio Build Number

_No response_

### Device form factor

_No response_

### Additional context

Related CommunityToolkit/Labs-Windows#13

### Help us help you

Yes, but only if others can assist.