In the US, there is not such a thing like ‘VAT invoice’ if I am not wrong?
Yes correct, there is no VAT invoice in US, if it’s an online order you usually get order confirmation, and order shipped emails. Sometimes a paper receipt is enclosed in the shipped order, and most of the times you can print/view receipts from merchant’s website if you have an account with them. If you do have an account with the merchant check in past orders.