Can I make this type of edit in Odoo SaaS?

Thanks.

If it's a technical customizations, you can modify the original "_name_search()" method, either by overriding or replacing it, as for example you can see as follows:

class ProductTemplate(models.Model):    
	​_inherit = 'product.template'    

	​@api.model    
	​def _name_search(self, name='', args=None, operator='ilike', limit=100, name_get_uid=None):        
	​	​args = list(args or [])        
	​	​if name :            
	​	​	​args += ['|', '|' , ('name', operator, name), ('default_code', operator, name), ('sale_desc', operator, name), ('prod_desc', operator, name)]        
	​	​	​return self._search(args, limit=limit, access_rights_uid=name_get_uid)

See better example from Cybrosis here: https://www.cybrosys.com/blog/how-to-create-name-search-function-in-odoo-17