The process is now usually handled online through the Office of Papal Charities' web site: https://www.elemosineria.va/
However, there's a declaration of eligibility of the person(s) for whom the Blessing is being requested.
With full responsibility and in right conscience that the person(s) for whom I am requesting an Apostolic Blessing
is/are living a Christian life in accordance with the faith and with Catholic morals;
does/do not belong to non-Catholic religious groups or associations or organizations that are publicly opposed to the Catholic Church;
does/do not cohabit; is/are not married exclusively civilly; or divorced and remarried exclusively civilly
is/are not under any canonical penalty legitimately applied or declared;
does/do not hold public office or public roles.
It is possible that by working through your local diocese/eparchy you might be able to get an exception to some of these rules. Requesting through the local diocese was t he process before this internet process was implemented.