Yes, this is possible by creating a Computed Field on the Request for quotation / Purchase Order.

My example is using the Customer Reference on a sales order, and assumes there's a one2one relationship (each PO is for one SO).  It should be easy to adapt for your custom field.

for record in self:
  if record.origin:
    order = record.env['sale.order'].search([('name','=',record.origin)])
    record['x_studio_client_order_ref'] = order.client_order_ref