First off, the mark of the beast will be put on people's foreheads or right hands:
And he shall make all, both little and great, rich and poor, freemen and bondmen, to have a character in their right hand, or on their foreheads. (Revelation 13:16).
You take the covid vaccine in your forearm, so it physically cannot be that.
If there was a vaccine that had to be taken in either the right hand or the forehead, and
if you were banned from buying or selling anything unless you had that vaccine, then I'd start talking about the mark of the beast. This isn't it.
Some people might argue that a future vaccine passport might be the mark of the beast, because some states are thinking about preventing people from going to the shops (buying) or working (selling) without a vaccine passport, and in these people's minds that passport will be in the form of a chip implanted in the hand or forehead. The problem with that is that hasn't happened. If a government required that, then I'd be concerned. They haven't.
For all this to happen, we need to be in the end times. This means that the Jews need to have converted to Christianity and recognise that Jesus is the Son of God:
And I will pour out upon the house of David, and upon the inhabitants of Jerusalem, the spirit of grace, and of prayers: and they shall look upon me, whom they have pierced: and they shall mourn for him as one mourneth for an only son, and they shall grieve over him, as the manner is to grieve for the death of the firstborn. (Zacharias 12:10)
Ask yourself; are the Jews about to "grieve over" Jesus? Do you really think that all 15 million Jews on this planet are about to start going to Mass?
Taking the mark of the beast would also be a mortal sin, as it would damn you to Hell:
The third angel followed them, saying with a loud voice: If any man shall adore the beast and his image, and receive his character in his forehead, or in his hand; He also shall drink of the wine of the wrath of God, which is mingled with pure wine in the cup of his wrath, and shall be tormented with fire and brimstone in the sight of the holy angels, and in the sight of the Lamb. And the smoke of their torments shall ascend up for ever and ever: neither have they rest day nor night, who have adored the beast, and his image, and whoever receiveth the character of his name. (Revelation 14:9-11)
Only mortal sins can damn a person to be "tormented with fire and brimstone", so taking the mark of the beast must be one. To commit a mortal sin you need to have full knowledge that what you're doing is sinful. (there are other requirements, but that's the most important one here.) We don't fully know that the vaccine is the mark of the beast; it isn't being advertised as a "mark of loyalty" to anyone, so full knowledge isn't present, meaning that taking it can't be mortal sin. This means that the vaccine can't cause you to be "tormented" and therefore that it can't be the mark.