How can I add Block Style support to the core HTML block in Gutenberg?

Block Style Variations allow providing alternative styles to existing blocks using filters, as explained here.

Here are some examples for your case using the server-side functions (php).
You can either add the css style inline:

register_block_style(
    'core/html',
    array(
        'name'         => 'full-width-video',
        'label'        => __( 'Full Width Video' ),
        'inline_style' => '.wp-block-html.is-style-full-width-video { width:100%; }',
    )
);

Or reference an enqueued stylesheet that contains that class

wp_register_style( 'myguten-style', get_template_directory_uri() . '/custom-style.css' );

// ...

register_block_style(
    'core/html',
    array(
        'name'         => 'full-width-video',
        'label'        => __( 'Full Width Video' ),
        'style_handle' => 'myguten-style',
    )
);

register_block_style should be hooked in an init action.