Andorra is outside the EU. The UK is currently inside the EU!
That means if you get something delivered to Andorra from the UK you don't have to pay the 20% UK VAT.
You will have to pay just 4.5% IGI in Andorra.
The simplest method I have found is to order the item and have it directly transported to Andorra.
Someone like the NA Grup (You see their trucks regularly around La Vella town) are good to the transportation.
http://nagrup.com/ Thomas there speaks English.
That way the selling company in the UK can simply not charge you any VAT on your Invoice as they have a delivery address outside the EU.
The delivery can get expensive, so you might consider doing it yourself.
In which case: The key phrase you are looking for the UK retailer is "VAT retail export scheme".
You won't be able to get anything by mail order, eBay or any other internet retailers doing it; you will have to pick up whatever you buying in person.
If they are member of this scheme you need to request the EU-Export under it.
They will give you an invoice that needs to be stamped by customs on the French or Spanish side.
You will then be able to use an agent (Credit Andorra Bank for example) to refund the VAT to you. They will make a hefty charge, so this really only works for higher value items.
Don't forget to stop on the Andorran side of the border and own up, and pay the 4.5% charge.