Short answer: No.
How can I be so sure? Because biblically speaking, the “Mark of the Beast” as noted in the Apocalypse of John (aka Revelation) is a way of declaring a higher allegiance to a power other than Jesus/God. It is not as if someone could slip “the mark of the beast” into your drink while you were not looking and then you become damned to hell by drinking that drink unaware. It is not a trick, though it is deceitful. It is taking what seems an easy way (perhaps under duress), or a powerful way (under temptation) that sets another authority above Jesus.
Let’s go with the drink example. Let’s say someone offers a toast to the emperor above all other kings and authorities, and you drink to that. Or what about offering a pledge of allegiance to a flag without “under God”? One could argue that this is much closer to taking the “mark of the beast” and certainly not an action a Christian should take. The mark need not be a tattooed “666” on your forehead, either. It is declaring loyalty to a power above your loyalty to Jesus.
So, if you had to sign an oath declaring your loyalty to a government or leader above all other powers including Jesus in order to take a dose of vaccine, then by all means, refuse the vaccine on those terms. To my knowledge, no one is requiring anything like this. Health officials and leaders in our society just want you and our community to live and be as healthy as possible.
You may wonder why I am even addressing this issue. It seems so obvious to most; am I serious? Well, this is apparently not obvious to everyone. I know this question has been asked earnestly by those who believe it to be so. So, just in case….