Father -in -law has trailers with no titles. He has been paying taxes on them, now he is very sick and we have no titles for the trailers. What can we do?

Your F-i-L should apply for a duplicate of the original title. You say he is very sick, so from an esate persepective it will be much easier if he gets this taken care of now rather than waiting for the executor of his estate to have to do it after he passes. The form for requesting a duplicate title is here:

NCDOT: Replacement & Duplicate Title

What kind of trailers? Depending on their weights there may not be titles.