The assertion is based on this verse.
Rev 13:16 -17 It (the Beast) also forced all people, great and small, rich and poor, free and slave, to receive a mark on their right hands or on their foreheads, so that they could not buy or sell unless they had the mark,
Between The Ears BLOG INDEX, with titles and summaries.
|Kanye West |
It is difficult to search the Internet for things that happen on social media. Youtube, Twitter, Facebook, and Instgram all have policies against misinformation. They can remove content they find offensive, leaving the trail of evidence impossible to follow. It's called Ephemeral Content . The Washington Post  article quoted above penetrates the social media fog better than I can.
But What Does The Bible Say?
Let's look more closely at what Revelation actually says about the Mark of the Beast
Revelation 13:15-18 11 Then I saw another beast coming up out of the earth; and he had two horns like a lamb, and he spoke as a dragon. 12 He exercises all the authority of the first beast in his presence. And he makes the earth and those who live on it worship the first beast, whose fatal wound was healed. 13 He performs great signs, so that he even makes fire come down out of the sky to the earth in the presence of people. 14 And he deceives those who live on the earth because of the signs which it was given him to perform in the presence of the beast, telling those who live on the earth to make an image to the beast who *had the wound of the sword and has come to life. 15 And it was given to him to give breath to the image of the beast, so that the image of the beast would even speak and cause all who do not worship the image of the beast to be killed. 16 And he causes all, the small and the great, the rich and the poor, and the free and the slaves, to be given a mark on their right hands or on their foreheads, 17 and he decrees that no one will be able to buy or to sell, except the one who has the mark, either the name of the beast or the number of his name. 18 Here is wisdom. Let him who has understanding calculate the number of the beast, for the number is that of a man; and his number is six hundred and sixty-six.
Mark on hand or forehead
Verse 16 says the mark, whatever it is, will be on a person's right hand or on their forehead. Vaccinations are given by injection in the upper arm. It just isn't given in the hand or the forehead.
Under Domitian (81-96 AD) in the time of this letter, emperor worship was compulsory for every Roman citizen. The burning of the pinch of incense was rewarded by a certificate which had to be renewed annually. Failure to produce a certificate meant being branded as Christian and this opened the way, if the magistrates so decided, for the death penalty on the ground of treason. - attributed to James Allen “What the Bible Teaches – Revelation” - That Pinch Of Incense 
|The Book of Revelation |
Jack Hibbs, a pastor, said in a sermon that he believes the COVID-19 vaccine is a social control tactic that is conditioning people to accept the mark of the beast. Belief Net Right-wing pastor Curt Landry posted a video on YouTube on Tuesday in which he told viewers never to accept any vaccine that is created to prevent the spread of the COVID-19 coronavirus because the vaccine will be a precursor to the biblical Mark of the Beast. Right Wing Watch The global society has to be conditioned in such a way as to submit to the mark, the beast, and the false prophet, in person or in proxy for the MARK to be effected. Duncan Mango 
The objective of such rules is nothing to do with public health; they are intended to exasperate us all to the point where we yield to the indignity of vaccine passports when they are rolled out, just so we can live our lives and be left in peace. Out Of Mind 
Digital-format immunity passports would eventually likely normalize digital-format proof-of-status documents.
Advocates of COVID passports visualize a world where we can’t pass through a door to a plane, workplace, school, or restaurant until the gatekeeper scans our credentials. In no time the public would be conditioned to submit to these demands.
This digital system could easily be expanded to check not just a person’s immunity status, but any other bit of personal information a gatekeeper might deem relevant, such as banking information, age, pregnancy, HIV status, or criminal history; and all data – your data – could be accumulated into one database. Clarion 
Without debate or democratic scrutiny, vaccine passports are quickly heading from unthinkable to unstoppable.Vaccine identity cards could be demanded ‘in hospitality settings’ — in other words, in pubs.As ever, in our new Orwellian world, the reverse language is used — restrictions will be lifted for those who comply. Spectator.co.uk 
The Mark Of The Lamb
Rev 14 Then I looked, and behold, the Lamb was standing on Mount Zion, and with Him 144,000 who had His name and the name of His Father written on their foreheads.
Ephesians 1:13 having also believed, you were sealed in Him with the Holy Spirit of the promise