Who owns the rights to a negative the purchaser of the photo or the studio?

The below does not constitute legal advice, does not form an attorney-client relationship, and should not be relied upon to take or refrain from taking any action.

Generally, the photographer or portrait studio retains the copyright in the photograph and the purchaser only owns the specific print(s) of the photograph taken. Some portrait studios will agree to assign the copyright rights to you or at least permit you to have access to the original photograph (be it film negative or digital file) for personal use, but this is not very common.

In short, the portrait studio is probably legally entitled to refuse to allow use of the negative, even under the circumstances you describe.

The family only bought the specific portrait, not the negative of the photo or the clopyright to reproduce the photo, so they have no legal claim to the negative. The family does have a pretty compelling moral claim to it, but the studio has the right to say no, as making copies is the studio's business and from their perspective, they know the family could afford the portrait in the first place.

Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.