I apologize if this post is too self promotional but we have a proprietary document retrieval service where we can get Cuban birth, death, marriage certificates along with school records and other documents.
We can get baptismal certificates too if we know the church where the person was baptized.
The only way to do a record search is to go to the civil registry covering the area where the person/family lived at the time.
There are no digital searches in Cuba.
We are not investigators but we have been fortunate to help many many people get the documents they need. In many cases we can find family members who are also registered at the same civil registry.