Can’t seem to filter wp_query by current category ID

You mentioned you are using a custom taxonomy – therefore including the cat key in your wp_query args won’t work. wp_query expects that to be the ID of a term belonging to the standard WP category taxonomy. With a custom tax you need to use the dedicated taxonomy parameters.

So your args should be something like:

$post_args = array(
    'post_type' => 'donor',
    'orderby'       => 'post_date',
    'posts_per_page'=> '-1', // overrides posts per page in theme settings
    'tax_query' => array(
        array(
            'taxonomy' => 'donor_taxonomy',
            'field'    => 'term_id',
            'terms'    => $category->term_id,
        ),
    )

);