Contact Form: improve security of the form endpoint (#39759)

* Contact Form: improve security of the form endpoint

Ensure that submitted forms can only be accessed by logged in users with the necessary capabilities.

Reference: p9dueE-8ng-p2

Co-authored-by:	Chris Jean <gaarai@gaarai.com>

* changelog

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Co-authored-by: Chris Jean <gaarai@gaarai.com>

Committed via a GitHub action: https://github.com/Automattic/jetpack/actions/runs/11323329319

Upstream-Ref: Automattic/jetpack@1a544bc