No, he did not.
Rudolf Joseph Lorenz Steiner (27 (or 25) February 1861 – 30 March 1925) was an Austrian, philosopher, social reformer, architect, esotericist and claimed clairvoyant.
Rudolf Steiner was indeed born a bit late in 1861 for having authored a seminal paper in the history of medicine and even virology in 1875 at the age of 14.
So, 'the doubting website' is correct in disputing even the possibility of Steiner being the author of this 'alleged' paper. But what's going on here?
Neither the CDC nor the NVIC cite any proper sources for their claims, one of which at least looks even like plagiarising the other. So, does any paper exist in the history of varicella research similar to the ones described above?
Going back we do find a few other papers that seem to make reference to the one sought:
- Steiner P. Zur inokulation der varicellen. Wein Med Wochenschr 1875; 25:306.
— C. Jo White: "Varicella-Zoster Virus Vaccine", Clinical Infectious Diseases, Volume 24, Issue 5, May 1997, Pages 753–763, 01 May 1997. https://doi.org/10.1093/clinids/24.5.753
But this is already a "P Steiner", not an "R Steiner".
Another reference —this time to "G Steiner"— is found in
1 Steiner G: Zur Inokulation der Varicellen. Wien Med Wochenschr 1875;25:306.
— Gerd Gross & Hans Wilhelm Doerr: "Herpes Zoster Recent Aspects of Diagnosis and Control", Monographs in Virology, Vol. 26, Karger: Basel, Freiburg, 2006, pIX.
Or
Steiner W. Zur Inokulation der Varicellen. Wien Med Wochenschr 1875:16:306.
— Robert Weibel et al.: "Live attenuated Varicella Virus Vaccine. Efficacy Trial in Healthy Children", New England Journal of Medicine, Vol 310, No 22, 31 May 1984, pp1409–1415. DOI: 10.1056/NEJM198405313102201
At other even earlier times this paper is referenced just as "Steiner" (doi).
Just like in the CDC's own earlier "Pink Book" editions, for example in 1996 or (3rd edition, 1996, p186; 6th edition, 2000, p171), which I could confirm up to 11th edition.
But perhaps as early as the 12th, quoted in question from the 13th edition, and even in the latest version, 14th edition from 2021, which claims to have zero errata pending, it says:
In 1875, Rudolf Steiner demonstrated that chickenpox was caused by an infectious agent by inoculating volunteers with the vesicular fluid from a patient with acute varicella.
Time to clear that up.
On that site claiming to be skeptical about "all of virology's lies", the very existence of the paper is called into question.
So, does this paper exist?
Yes. It does. Although for example PubMed is unaware of it.
It is shown below, and thus any further speculation arising from 'non existent paper' seem moot…
The references uncovered so far are sometimes quite a bit wobbly on the details.
The proper citation is spelled out in full:
For the paper, that one is called: "Wiener Medizinische Wochenschrift", (not something like "We in" etc) which is archived and digitized at the Austrian National Library
Since the article is very short, I reproduce it here in full in two pictures:

That is:
— Prof Steiner (Vorstand der pädiatrischen Klinik zu Prag): "Zur Inokulation der Varicellen", Wiener Medizinische Wochenschrift, Vol 25, Nr 16, p305–308, 17 April 1875. (direct-link)
As can be seen, the columns are the page numbers, so a proper citation would list 305 as the reference, not 306 as the examples above.
The author is now just given as "Prof Steiner of Prague". Absolutely no first name present on that paper.
Even assuming a "Genius Rudolf Steiner", writing about an experiment at 14 years of age might be remarkable, but holding the title of Professor seems impossible.
So who is this Steiner, if not a Rudolf?
We find for example another standard work of reference from that time, a book called "Compendium der Kinderkrankheiten für Studirende und Ärzte", authored by Johann Steiner, Leipzig: Verlag von F.C.W. Vogel 1872. From this work is its English translation on archive.org:
FIFTEEN years. of uninterrupted activity in the Francis
Joseph Hospital for Children in Prague, […] and partly in the
independent position of Teacher and Physician in Ordinary to the Hospital, have encouraged me and given me some claim to write
this treatise. […]
- Varicella, chicken-pox:
This is an acute, epidemic, and
— from "Author's Preface" in the English version from 1875, (preface unpaginated, p11 in PDF; but on the topic of varicella: p351-358)
This Johann Steiner led the Children's Hospital in Prague ("Franz-Josef-Kinderspital") at the time, and has its own biography entry read:
Steiner, Johann (1833–1876), Pädiater und Fachschriftsteller
1866 ao. Prof. für Kinderheilkde., führte er ab 1874 die in dem Kinderkrankenhaus errichtete Kinderklinik. Wiss. befaßte sich S. in erster Linie mit verschiedenen Kinderkrankheiten, insbes. mit den Varicellen und deren Impfbarkeit, […]
This addition of an arbitrary erroneous first name seems to have happened around 2008, with — Gretchen Hoffmann: "Chickenpox", 2008. among the first to conjure it up, at least as far as full text searches of digitized books available via net searches allow such conclusions.
Curiously, even though a 2010 German text book still uses only the last name in its 2010 third edition, when this very same edition was translated into English, the name "Rudolf" appears.
(German: — Susanne Modrow, Dietrich Falke, Uwe Truyen, Hermann Schätzl: "Molekulare Virologie" (3rd German edition), Spektrum: Heidelberg, 2010, p588. worldcat Versus: — S Modrow et al.: "Molecular Virology" (1st English edition), Springer: Heidelberg New York, 2013, p798. worldcat)
Conclusion
No, Rudolf Steiner did not conduct these experiments or write anything in 1875 about varicella viruses. Johann Steiner did.
All those websites —like the CDC in their reference 'pink book', and those then copying and thus propagating the errors from it — use an unverified, unreferenced and very possibly plagiarized invention for the first name that is missing in the original publication.
As listed in examples above: Many authors accumulated different errors (plural!) in their copying of personally unread reference material: for the author names, the article titles, the journal names, the page numbers, in essence pretending proper reference work, and thus committing plagiarism in that part.