No, It includes the VAT already paid to HMRC through the materials supplier!the £262 charge does NOT include VAT to be paid to HMRC
... and I don't care how much you shout or protest, your supply - or any other supply (of the type under discussion) - is NOT VAT free.
Food isn't VAT free even though some of it is Zero Rate, VAT has to be accounted for even though the rate is 0% but there I am being pedantic
There are two categories of supply which are VAT Free - one is 'Exempt' and the other is 'Outside Scope' - neither of which apply in your case.
Postage Charges are a peculiarity since they are 'Exempt' - ie. you don't pay VAT on stamps - but if they are shown separately on an invoice from a Registered supplier then they attract the 'Standard' rate.