It's difficult to document that a law requiring a person who cannot obtain proof of birth in the US to register his/her birth does not exist, since there is such a vast volume of state and federal law. But a document from USCIS states
If you were born in the United States, you do not need to apply to
USCIS for any evidence of citizenship. Your birth certificate issued
where you were born is proof of your citizenship.
There is a footnote explaining this does not necessarily apply to children of foreign diplomats.
So if there is a law, it would have to be a West Virginia law, and I am not aware of such a law. If such a law does exist, and violation of the law is a crime, then the State of West Virginia would have to prove all the elements of the crime beyond a reasonable doubt.
The first element would be that the person was born in West Virginia. If proof beyond a reasonable doubt exists, and is in the possession of the prosecutor, no doubt the person's lawyer could find a way to compel the state to perform the necessary registration on the basis of the evidence the state already possesses, and the case becomes moot.
Another element that is likely to be in the law is that the person knows he/she was born in West Virginia. But if there isn't enough evidence to complete the registration process, the person could argue that he/she does not know if he/she were born in West Virginia.
If the person works, it is likely the person will appear to violate tax laws, because even if income taxes are withheld from wages, with no SSN to connect the person to the money that was withheld, it will appear the person failed to pay taxes.
In response to a comment by ohwilleke I will show why it was never the responsibility of the person who was born to file a birth certificate. The relevant WV law states
(a) A certificate of birth for each live birth which occurs in this
state shall be filed with the section of vital statistics...
[if the birth is outside an institution the birth shall be reported
(1) The physician in attendance at or immediately after the birth;
(2) Any other person in attendance at or immediately after the birth;
(3) The father or the mother, or, in the absence of the father and the
inability of the mother, the person in charge of the premises where
the birth occurred; or
(4) Any other person qualified by the department by rule to establish
the facts of birth.
I think it would be absurd to interpret the newborn as being in attendance at his/her own birth, and equally absurd to consider him/her to be qualified to establish the facts of birth.