Add ability to send emails from JSON:API contact form.

[Decipher]

Is your feature request related to a problem? Please describe.
Currently the Contact module only sends mail when submitted via the Drupal form, not when submitted via the JSON:API.

Describe the solution you'd like
Allow JSON:API contact form to send mail.

Describe alternatives you've considered
N/A

Additional context
https://www.drupal.org/project/drupal/issues/2936644