I think its theologically possible for lay people, including women, to be advisors to the Pope, hold bureaucratic positions in the Curia, and even be electors of the Pope. However, I’m not so sure it is possible to make them Cardinals, at least how the office has been traditionally defined. Cardinals are by their nature clergy (in the narrow sense of the word). Even “lay cardinals” in the past were not actually laymen, strictly speaking. They always received the tonsure and were in the minor orders.
The cardinalate is not just the electoral college of the papacy or advisors to the Pope or even just an honorary title. The title of Cardinal grew out of their clerical status. It was originally a title given to all clergy permanently attached to a church/diocese, or “incardinated” as we say today. Then it began to be reserved to those in prominent Sees, then to the chief clergy there, and finally to those of the Roman See only–kind of like how the title “Pope” became more specific over time. Just as when various bishops had the title of Pope, the Bishop of Rome was still the head, the status of the Roman clergy was always special. As the clergy of the chief particular Church, they were given special honor–which is why even in the very nascent Church during an interregnum, difficult questions were submitted still to the Roman priests (since the Roman Church was the head of all the other Churches).
To this day Cardinals are clergy of Rome. When clergy outside Rome are made Cardinals, those clergy are given parishes in Rome. For example, Cardinal Dolan of New York presides over Our Lady of Guadalupe at Monte Mario in Rome or Cardinal DiNardo of Houston, Texas presides over St. Eusebius Church in Rome. Yes, the College of Cardinals as an electorate is only 1000 years old, but their office as Roman clergy goes back to the beginning.
Seeing the cardinalate as merely bureaucracy or even the electors of the Pope, misses the point. Lay people in the Roman bureaucracy (always subject to the bishop, in this case, the Pope), is fine. Lay people electors of the Pope is also theological fine–the election of a Pope can happen however the previous supreme authority legislates (it is very common for a sitting Pope to slightly modify the election law). In the past, lay emperors were given a vote or veto power or even the only say; theoretically other lay people could be given a vote, or the sitting Pope could choose his successor, etc.
But bestowing the cardinalate on such lay bureaucrats or electors would deviate from what a cardinal truly is, which is clergy. Lay emperors and whatnot who voted in papal elections, were not made Cardinals, for example.
Sure, the word “clergy” can be used in a very broad sense to include abbesses and nuns, and really anyone involved in any service of the Church (even lay folks who begged on behalf of convents, etc.) and abbesses do have a kind of jurisdiction since they have persons subject to them. But the minor orders were different. When Paul VI abolished them, he maintained instituted Acolytes and Lectors, but explicitly reserved them only to men. Paul VI (and later John Paul II) seemed open to, and pretty much did, change everything that could possibly be changed, and yet the instituted ministries were still reserved for men alone, even when temporarily deputized (at least de jure) lay readers, altar servers, EMHCs could all be women. The reasoning I can find for this is that the minor orders while merely sacramental, were derived from and a special participation in the sacrament of holy orders–and this was the reasoning behind doing the same for the instituted ministries, despite the fact that they are much less of a participation in holy orders than the minor orders were (I think it’s probably not definiively settled whether or not they are far enough removed to be open to women). I know the question of the possibility of admitting women to the instituted ministry of lector was suggested at a previous synod of bishops and the Pope did not do so in the resulting Apostolic Exhortation (interestingly enough, some Eastern Orthodox Churches have begun tonsuring women as lectors from what I’ve read).
So my argument would be that the office we call “Cardinal” is so closely tied to the sacrament of Holy Orders that women could not be admitted to them. I guess you could let people occupying a different office, wear a red hat, and be called a “Cardinal” (it’s just a word after all), but I don’t think a woman or layman could hold the office we have denoted by the word “Cardinal” for over 1000 years.
