If you want to call the shortcode as AJAX, then with the standard WordPress calls, you can’t make use of the do_shortcode() function however you can call the class that builds the shortcode directly.
Note: if you want to use popups and ajax life become a bit more complex as you will need to load the Eventbrite popup scripts too. If you want to do this you may want to ask for directions.
In this example, I have built a search filter and button, of course you can do what you like, such as dynamic tabs. The example works and is visible on the demo site here, however little attention has been paid to finessing the style etc, as it is expected you are a developer if you are reading this, so you will make it your own.
This is written as a plugin, however, with a few tweaks you could use it in the functions.php for your child theme.
First we will create the html mark up for the search box and results area ( you can put this in page or post for example ). I have kept this simple
The structure of our plugin directory in this example is:
test-ajax | -- test.js -- test-ajax.php
You can use what you like, or as mentioned add code to your child theme. If you are building it as a plugin, once created you can zip the directory up an upload and activate as you would any plugin.
Our PHP code to enqueues the jQuery , add the ajax actions and the function to get the filter and display output
You could easily extend this for instance to switch between list and grid layouts etc.
Full description of making ajax calls in WordPress can be found in the WordPress Developer Codex.