I’m afraid acf_form() is designed to redirect the user after the form is successfully submitted. If you need to do it using AJAX, then you need to create a handler to handle the submitted data. This page should give you more idea about it: https://codex.wordpress.org/AJAX_in_Plugins.
In the custom handler, you can use the update_field() function to add the value to the post.
I hope this makes sense 🙂
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