If lastminute.com take a commission from the hotel, rather than buying a room at a discounted rate and then charging a higher rate to you, then VAT is not chargable on a commission payment.
Therefore it would mean they may not need to be VAT registered – hence unable to issue a VAT receipt!?! But if that was the case the hotel should be able to issue a VAT receipt with no issue as it would show the full payment for the room and correct VAT amount.
All seems very odd either way…maybe try asking the nice (!?!) VAT folks direct?