Hi Ken, Yes this is a UK site, but there are members worldwide and forums that cover the USA, so we should be able to dig something up for you.
If a Yearsley family member happens to search for a family name they will probably find this thread, so even if we can't find a direct descendant, they may still find you one day.
We often have challenges like this on rootschat to reunite family with precious items. I'm sure the photographs will be especially interesting for anyone with connections to the Yearsleys.
If you are unable to decipher some of the writing, you can always post the relevant bits here for us to have a try.
Very unusual middle name Ten Eyck. It has got me curious.

So William TE Yearsley is the earliest birth in this book at 1844? As you say, it appears to be an English surname, so it will be interesting to work out who was the first Yearsley to come to the USA.
I know you have posted this in the correct forum (Artifacts/Family Bibles) but you may get a better response and forum members with more knowledge of how to research this family if this was on the USA forum. I will leave it to a moderator to decide.
**It's a good thing that you are doing to try to reunite this bible with family. I wish you luck.
* Moderator comment: topic moved to USA board - I agree that it would seem most suited there
*