Yes, well, there hasn't been a specifically named, "Covid emergency declaration", to my knowledge either. You are correct regarding the complete, (or partial), expiration of Executive Order 10294 in May. My bad, I had the wrong M-month in my head when I typed the query into my phone earlier this morning.
Regarding Federal authority to enforce and require a Covid-19 vaccination among foreign immigrants.
There are several executive orders going back to 2014 that pertain. Among them are;
13295 - https://www.govinfo.gov/link/cpd/executiveorder/13295
13375 - https://www.govinfo.gov/app/details/CFR-2006-title3-vol1/CFR-2006-title3-vol1-eo13375
13674 - https://www.govinfo.gov/app/details/CFR-2015-title3-vol1/CFR-2015-title3-vol1-eo13674
13674 is very large. I am currently reading to find how it pertains to a vaccination requirement. In or around page 498 of the section titled Subchapter F, Part 70 of the Section titled Title 42-Public Health, the section gives authority to the CDC to determine what constitutes a, "quarantinable communicable disease."
Searching the CDC database I found this page;
https://www.cdc.gov/immigrantrefugeehealth/panel-physicians/covid-19-technical-instructions.html
Also, this CDC page references these Executive orders and also Congressional Code title 42;
https://www.govinfo.gov/content/pkg/USCODE-2011-title42/html/USCODE-2011-title42-chap6A-subchapII-partG.htm
The CDC posted above states therein, (I highlighted the important part);
"Applicants*, defined in these Technical Instructions as people applying for immigrant or refugee status, as well as non-immigrants (NIVs) who are required to have an overseas medical examination, are medically screened days or weeks prior to travel to the United States (US). Thus, a negative screening for COVID-19 at the time of the medical evaluation does not guarantee the applicant will not have COVID-19 at the time of immigration to the United States.
A combination of vaccination, strategic testing, and routine infection control practices will provide the best protection from COVID-19 for applicants and US communities. These instructions provide requirements for COVID-19 vaccination and testing for applicants. The Instructions in this document are to be followed for COVID-19 when assessing applicants from all countries. These Technical Instructions became effective October 1, 2021, remain in effect, and may be updated periodically.
Visit the Technical Instructions for Panel Physicians webpage for more information about the medical examination for applicants for US immigration.
*Refugees and NIVs, including K-visa applicants are not required to receive the COVID-19 vaccine as part of the immigration medical examination; however, the other components of these Technical Instructions do apply to all applicants."
It would seem however, that the code allows exception for all K-Visa applicants however I haven't tracked down the specific subsection of code within Title 42 where this is referenced. (Going by the CDC blurb above.)
All said, as we have seen in the news over the past year+, there is a move on to axe Title 42. The current administration has come very close, albeit presently in court.
Should Title 42 suddenly disappear, it may also be that Federal authority to enforce any vaccination originating from a CDC declaration of a, "quarantinable communicable disease" will go with it. Although I'd assumed there will be something to replace it.
Regarding the other piece of the puzzle, travel requirements for Foreigners to enter the US;
Executive Proclamation 10294 of October 25, 2021 - Advancing the Safe Resumption of Global Travel During the Covid-19 Pandemic. It may be referenced in the Federal registry here;
https://www.federalregister.gov/documents/2021/10/28/2021-23645/advancing-the-safe-resumption-of-global-travel-during-the-covid-19-pandemic
Some of this has been rescinded by CDC Proclamation, i.e. dumping testing requirement to fly, getting rid of vax requirement entering by land ;
https://www.cdc.gov/quarantine/fr-proof-negative-test.html
https://www.federalregister.gov/d/2022-01402
https://www.federalregister.gov/d/2022-01403
So long story short...
I do not believe foreign applicants are required by US law to be vaccinated against Covid as a requirement, qualification for a K-Visa nor to enter the US via a land border.
I am continuing to research for more clarification, the specific section within title 42 pertaining to the exemption for "Refugees and NIVs, including K-visa applicants". It is understandable that personnel at embassy level may not be aware of the exemption for K-Visa applicants. They are likely certainly aware of asylum seekers, refugees, and NIVs exclusion status. This jibes with accounts I have read here where some say it's a requirement and others say it isn't. Understandable as the vast majority will travel by air thus required vaxxed under travel policy above.
Would be great to get more opinions, experiences, etc... and nail it all down. Anyone who has first hand knowledge please post it up!