Can I claim VAT back on my mileage?

If your company is on Standard VAT then yes.

You can do this in your mileage log by selecting your vehicle’s engine type against a mileage entry. This applies to all car and motorcycle journeys made. Going forward, all new entries made will automatically have your vehicle’s engine type set. This will then be added to your next VAT return to claim back VAT using the HMRC Advisory fuel rates for that period. 

Don’t forget that with any VAT claim you should have a corresponding receipt with enough VAT to cover the claim, this includes any fuel receipts.