Post not populating for custom post type based on category selection

But this part is not executing →

$args = array(
    'category_name' => $taxonomy,
    'post_type' => 'video',
    'posts_per_page' => -1,

It probably is. It’s just wrong.

If you want to query posts by terms in a custom taxonomy you need to add a taxonomy query:

$args = array(
    'post_type'      => 'video',
    'posts_per_page' => -1,
    'tax_query'      => array(
        array(
            'taxonomy' => 'vcategory',
            'field'    => 'slug',
            'terms'    => $taxonomy,
        )
    )
);

This is explained, with examples, in the documentation. I’m going to implore you again to please read documentation.

Also, you’re using the wrong $_POST field. In wp_dropdown_categories() you’ve given the field the name categoryfilter, but are trying to get the value for taxonomy. You need to set $taxonomy like this:

$taxonomy = $_POST['categoryfilter];

Also, you shouldn’t be using POST for this type of request. You should be using GET:

GET is used to request data from a specified resource.

POST is used to send data to a server to create/update a resource.

— https://www.w3schools.com/tags/ref_httpmethods.asp

You’re doing the first thing.