How can I customize the vendor page template?
If you want to customize the vendor page via HTML or CSS, the first thing you need to know is that the base template used by vendors is archive-product.php, which is the same one used on the Shop page.
This is why all of the changes you are going to make on this template will be applied to the store as well.
if you wish to use a different template, the solution would be to create it inside your theme and, using the template_include filter, let it load when the displayed page is a taxonomy page of the yith_shop_vendor kind.
There is an alternative solution to get to the same result: using the WooCommerce template hierarchy. WooCommerce allows you to use custom templates specially created for taxonomies.
To do this, you need to find a file named taxonomy-yith_shop_vendor.php and copy it inside the WooCommerce folder in your theme.
In this case, the plugin will use this template instead of the default archive-products.php one.