[TASK] Use AJAX API in FormEngine

This patch replaces the usages of `$.ajax()` and its friends with the
AJAX API provided by TYPO3 Core.

Resolves: #90038
Releases: master
Change-Id: Ief3767dd5a5256dca269b8dfb496f37a9bcdbe69
Reviewed-on: https://review.typo3.org/c/Packages/TYPO3.CMS/+/62790


Tested-by: TYPO3com <noreply@typo3.com>
Tested-by: Jörg Bösche <typo3@joergboesche.de>
Tested-by: Steffen Frese <steffenf14@gmail.com>
Tested-by: Henning Liebe <h.liebe@neusta.de>
Tested-by: Tobi Kretschmann <tobi@tobishome.de>
Reviewed-by: Jörg Bösche <typo3@joergboesche.de>
Reviewed-by: Steffen Frese <steffenf14@gmail.com>
Reviewed-by: Henning Liebe <h.liebe@neusta.de>
Reviewed-by: Tobi Kretschmann <tobi@tobishome.de>