If you receive a VAT invoice for booking agent commissions from a UK based agent it will itemise the VAT you have been charged and, provided you are VAT registered, you will be able to claim the VAT back, thereby reducing your overall VAT bill,.
If the booking agent is not based in the UK (for example, Booking.com) you may still get a ‘VAT invoice’ but it will not itemise VAT, and the cost cannot reduce your VAT bill (because no VAT has been charged).
The reason a VAT registered business is not charging you any VAT is the ‘reverse charge’ rule, which aims to simplify EU trading by moving the responsibility for reporting inter-EU transactions from the seller to the buyer of goods or services, thereby reducing the need for sellers to register for VAT in the country where the supply is made.
Ordinarily, where the ‘reverse charge’ applies, the recipient of the goods or services reports both their own purchase and the supplier’s sale in full on their own VAT return. These two declarations on the one VAT return is cash neutral but records the fact that a VAT transaction has taken place, so it is visible to the tax authorities.
However, if you make ‘partially exempt’ supplies, you may not be able to claim the whole of the ‘notional’ input tax back and that could result in a purchase actually increasing your VAT liability, rather than reducing it as is normally the case!
Perhaps unsurprisingly, it is not uncommon to see businesses getting this wrong, and claiming the input tax on invoices from booking.com etc, whilst not declaring the notional receipt to negate that claim. Making that mistake can cost you dearly in penalties and interest, so if in doubt ask!