Is this the only access to your property? If so it seems very strange it has not been picked up on before now, when you bought it was it not mentioned then?
Has your solicitor gone through the deeds with you and explained exactly what the issue/error is and why, when you believe there is a right in the deeds that there now appears not to be one eg a personal right granted which has now lapsed or maybe the right being misinterpreted?
I would assume the property is now registered as you acquired the property 12 years ago. It may, if there is a right in the deeds, be worth getting your solicitor to check with the Land Registry about whether or not rights exist.
An indemnity policy will only give an indemnity fund for certain uses such as possibly legal action if there is a problem eg the owner of the land blocking the access or maybe to help negotiate a right (which can be horribly expensive depending on who you're negotiating with).
There will be conditions attached to the policy to satisfy before it is put in place and others to adhere to after eg the indemnity insurers are probably asking for a statutory declaration (a statement sworn before a solicitor - criminal offence to swear a false statement) confirming uninterrupted use without consent, payment, problem etc... to back up your claim.
Other conditions can include that no approaches should be made to any third parties, such s your neighbour or the person who owns the access if different. This is to protect the insurers against a greater likelihood of a claim if other parties are made aware and try to assert rights such as blocking the access. In this case such an approach would be likely to void the policy so be careful who you talk to.
It might also be worth checking whether the proposed policy has been approved by the buyers solicitors and their lenders as, depending on the circumstances, this may cause a snag