Does the Catholic Church have an official position on domestic 'health passes' or 'vaccine passes' that could be used to control citizens' activities?

As far as I am aware, the answer to the particular question is no. Generally, the Catholic Church would assert that government has no right to infringe on a man's conscience, but that still is subservient to the common good (for instance, the government doesn't have a right to compel you to go to Mass if your conscience convicts you that the Roman Catholic Church is the who*e of babylon or something like that, but it does have the right to stop you from killing someone, even if your conscience convicts you that you have a duty to kill this person).

It will ultimately come down to a matter of prudential judgement. If you believe that the good of a 100% vaccinated population is important enough to take precedence over man's conscience in this case, then you will interpret the teachings of the Church to allow for vaccine passports (though in no wise to compel governments to implement them). If you do not think that COVID-19 is serious enough to warrant such draconian measures, you will interpret Church teachings to be against mandated vaccines.

Edit: As eques points out below, it's not right to say that one would interpret the Church's teachings to permit or forbid this. Rather, what the Church does teach is that one must evaluate for himself whether the conditions above pertain to the question. A government could coerce or not on this, and whether they ought is a matter of prudence. On my, and I suspect many others' reading of the teaching on conscience, the Church encourages governments to err on the side of permitting their citizens to follow conscience freely.

More on Catholic Conscience; the section on coercion may be especially helpful.

From the National Catholic Bioethics center, which is neither official church teaching, nor a conspiracy website but probably the closest place we can get unbiased (polemically at least) advice for Catholic ethical quandaries:

The Catholic perspective on the problems with requiring a COVID-19 vaccine passport looks first at the fundamental liberties of persons. The Church is calling people to make a careful discernment in conscience regarding taking a COVID-19 vaccine or not. This means that individuals have a strong right to be free of coercion to take a COVID-19 vaccine. They should also not be prevented from getting vaccinated if they qualify for ethically distributed vaccines and have made a well-considered decision to go forward with it. The proposals for the use of new vaccine passports that would involve discrimination against persons who choose not to accept the COVID-19 vaccines must be opposed by Catholics.

National Catholic Bioethics Center

The note from last December about getting the vaccine is not at all saying one must be vaccinated. Basically, Catholic ethics says

A human being must always obey the certain judgment of his conscience. If he were deliberately to act against it, he would condemn himself. Yet it can happen that moral conscience remains in ignorance and makes erroneous judgments about acts to be performed or already committed.

CCC 1790

The problem, that the consciences of individuals are trampled on by the state is not one to take lightly. The Catholic Church says we're supposed to follow our conscience, even if it sucks! Society is constantly telling us, especially children, to "be yourself", the Catholic Church says we should think for ourselves. And that's a remarkable thing, when she's the arbiter of faith and morals.

From the December note, which you'll probably see referenced, by way of the New York Times lens, would have your employer, even a Catholic hospital, believe that because vaccines are licit, that they're OK to mandate, but another equally understandable way to read what the vatican is saying is. The vaccines are licit, so it's OK to take them, but they're only barely OK and should never be mandated because they're so flawed.

And above that, I've never seen the Catholic Church have a take on the nature of the Vaccines available in America and Europe, the mRNA vaccines, which are transhumanist in nature, according to Elon Musk at least.

The Catholic Bioethics Center, does take on that argument though:

Myth 1: For vaccines that rely on injecting patients with mRNA, the possi­ble incorporation of these genes into our genetic makeup will fundamentally alter who we are as humans, moving us into a project of Transhumanism, the production of a “Human 2.0,” etc.

Reply: Any incorporation of new genes into our chromosomes from a COVID-19 mRNA vaccine would be an exceedingly rare occur­rence, if it were to occur at all. It is actually very difficult to get the ge­netic information of mRNA to in­tegrate into our chromosomes, partly because this would mean a reverse directional flow of the so-called Central Dogma of Molecular Biol­ogy: our DNA or chromosomes are read (“transcribed”) to produce mRNA, which is then read (“trans­lated”) to make proteins. Even if the accidental and unintentional incorporation of an mRNA mes­sage into our chromosomes were somehow to occur following vaccination, this would not mean that we were creating “Human 2.0,” since those genetic changes would not be expected to affect our sex cells, and therefore would not be transmitted to the next generation. Vaccinating people with an mRNA vaccine for COVID-19, therefore, does not imply that we are “remaking man” or heading down the path of Transhumanism.

https://www.ncbcenter.org/making-sense-of-bioethics-cms/column-182-covid-19-vaccine-myths

But even this, depends on a very tight scope of transhumanism. Viral vector vaccines in the first place, were transhumanist in nature, but relied on natural processes. One could argue that all medicine is transhumanist, to some extent. But mRNA vaccines are clearly a vector for transhumanism, in the form of mRNA manipulation, to take root in the world. It has moral implications, whether individual Bishops or the Pope realizes it at this moment or not.

The Church already gives us a clear picture of what we should expect from science when considering IVF technology:

It would on the one hand be illusory to claim that scientific research and its applications are morally neutral; on the other hand one cannot derive criteria for guidance from mere technical efficiency, from research's possible usefulness to some at the expense of others, or, worse still, from prevailing ideologies. Thus science and technology require, for their own intrinsic meaning, an unconditional respect for the fundamental criteria of the moral law: that is to say, they must be at the service of the human person, of his inalienable rights and his true and integral good according to the design and will of God. The rapid development of technological discoveries gives greater urgency to this need to respect the criteria just mentioned: science without conscience can only lead to man's ruin. "Our era needs such wisdom more than bygone ages if the discoveries made by man are to be further humanized. For the future of the world stands in peril unless wiser people are forthcoming"

https://www.vatican.va/roman_curia/congregations/cfaith/documents/rc_con_cfaith_doc_19870222_respect-for-human-life_en.html

In summary the wisdom of the ages is being replaced by the state (who tells you to only listen to the state media), business (who fires you or prohibits your access to goods) and science (who quash any opposition to the status quo) teaming up to force us to carry passports of a vaccine that clearly violates our consciences. But the Church says we should follow our consciences, even if nobody is doing it, even if it hurts us and even if it's wrong.