In the UK
This should be fairly easy to acquire unless you are domiciled outside of Europe. One of the few companies who cover imported vehicles as well as non-UK drivers is HIC who offer 'banger rally' insurance. Because this is so specialised depending on vehicle and personal details it is recommended to contact them to check eligibility and prices. It also may be worth checking your existing insurance policy to see if it offers cover in the UK and Europe
All UK insurance policies covering UK registered vehicles also provide the minimum third party cover in other EU countries as long as you tell the insurer you are going to be driving in those countries. However, you need to be 100% clear and honest with an insurer regarding how you’re using your vehicle, and that means being clear with them that you are taking part in the Mongol Rally if they ask for such information.
Unfortunately here it can get a little tricky. Sure, the Mongol Rally is not a race, and the vehicles must be road legal, so you ought not to be exposed to any more risk than you would on any other road trip through Europe, but being part of an organised event would make many policies void. A void policy can have large financial and legal consequences if you find yourself in an accident.
Finding an insurer who is happy to cover you for the Mongol Rally has been incredibly difficult for some teams in the past. One company that has helped again is Herts Insurance ‘banger rally’ policy.
If your vehicle is not registered in the EU then you will either need to find an insurer from your domicile that will be happy to cover you driving your vehicle in other parts of the world or find a local insurer that can cover your vehicle