Triweb Digital

This plugin injects a small search box at the top of taxonomy metaboxes (the checkbox lists) on the post edit screen. It only appears for taxonomies that have more than 5 terms to avoid clutter.
Search queries are performed via AJAX and the checklist is filtered client-side for a quick, responsive experience.

Taxonomy Admin Search

Adds an AJAX-powered search box to taxonomy metaboxes on the post edit screen when a taxonomy has more than 5 terms.

Shahala Anjum

With This plugin you can easily create a Ajax Search and Filter For Custom Posts and Any One Custom Taxonomy, once you install and activate the plugin, you can use the below Shortcode to display the filter and posts.
Display Default posts with category filter :
[ajax_posts_filter]
Display any custom posts types with any custom taxonomy with attributes as follows:
[ajax_posts_filter postype=”YOUR_CPT_NAME” taxtype=”YOUR_TAXONOMY_NAME”]
Currently The plugin supports one post type with one taxonomy.

Ajax CPT and Taxonomy Filter

Handle Ajax Search and Filter For Custom Posts and Any One Custom Taxonomy

Taxonomy Admin Search vs Ajax CPT and Taxonomy Filter