Hi,

You can add filter by inherit view of product template search view

<?xml version="1.0" encoding="utf-8"?>
<odoo>
    <data>
    	 <record id="product_template_search_view_inherit" model="ir.ui.view">
            <field name="name">product.template.search.inherit</field>
            <field name="model">product.template</field>
            <field name="inherit_id" ref="product.product_template_search_view" />
            <field name="arch" type="xml">
            <xpath expr="//filter[@name='inactive']" position="after">
	              <filter string="Vendor Product Name" name="vendor_product_name" domain="[('seller_ids.product_name','!=',False)]"/>
             </xpath>
            </field>
        </record>
    </data>
</odoo>

Thanks,