Im buying a shop as an investment. It will probably be let out.

My main business (unconnected with this purchase) is VAT exempt due to the nature of our service.

The shop is brand new so VAT is levied.

How can I claim this back?

It's really as simple as registering with HM Customs VAT and reclaiming it.Call yourself "Dogbox Enterprise" or something and register.
Of course,you will have to pay VAT on anything that your business sells,but I expect that will be zero.(Rent doesn't count)
Remember to apply for monthly VAT payments rather than 3 monthly,so as to get your money back earlier.

Instruct your solicitor treat the transaction as a Transfer of a Going Concern (TOGC), if possible. This will avoid the VAT issue altogether.