I want to take my dog to Fueteventura for my 10 day holiday in December, will this be possible?

Try contacting them and asking.

Talk to your vet. You need a pet passport and anti-rabies injections 6 months before you travel, so the answer to your question is NO.

You also need to consider the breed and the weather it will have to endure on the islands.

Not if the time limit for vaccinations and pet passport have passed. For just 10 days, it could cost more than your holiday.

https://www.gov.uk/take-pet-abroad/overview

Spain, one of the Canary Islands.

Where is Fueteventura?