The cart is hooked in using the storefront_header action. To remove it:

/**
 * Remove the cart from the theme header.
 */
function prefix_storefront_mods() {
    remove_action( 'storefront_header', 'storefront_header_cart', 60 );
}
add_action( 'init', 'prefix_storefront_mods' );

If you’d like to remove it from just the home page (or other specific pages), you can use a conditional statement:

function prefix_storefront_mods() {
	if ( is_front_page() ) {
    	remove_action( 'storefront_header', 'storefront_header_cart', 60 );
	}
}
add_action( 'init', 'prefix_storefront_mods' );