VAT is only claimable by VAT registered businesses and individuals, who can reclaim VAT charged to them on inputs (purchases) that they use in the course of trading. They then have to charge VAT on all their outputs (sales).
Government departments can only claim/charge VAT on items bought and sold in the course of trading activities.
I'm therefore fairly certain that the MoD can not reclaim VAT for normal expenses claims, though if the MoD supplies a service (band, aircraft for films, etc) then the MoD could reclaim the VAT element of costs incurred, but would have to charge VAT on the service.
Quite why a VAT receipt is needed by other depts. is unclear, but I'm certain that you can't personally reclaim any VAT, otherwise you'd be getting back more than you paid out, surely?