I mean, you can, but the cases in the US were imported from elsewhere: the ones in the Northeast are generally from heavily Orthodox Jews traveling to or from Israel; the cases in the southwest started among the Mennonite community and was imported from Mennonites in Mexico. Unfortunately, vaccine resistance is pretty high among both groups.
The better way to handle it would be to require that travelers to your country have been vaccinated, and that refugees and asylum seekers are kept in quarantine and given vaccinations before they're paroled to the general public. There'll still be people who slip through the cracks - immigrants who manage to avoid processing, people traveling on small boats or small planes, etc - but it would be a start. Unfortunately, I don't know how well that plays with all the various "free travel" rules.
You know how there's always this one lingering childhood resentment that's hard to let go of? Here's mine:
I don't remember exactly how old I was, just that it was some age when candy and presents were really important. But however old I was at the time, I got chicken pox as a kid (no vaccines for it back then).
I got it around Halloween, and my parents wouldn't let me go out trick-or-treating. I objected strongly because, you know, candy! So they sent my sisters out with a spare pillowcase to collect candy for me, which they dutifully did.
When they got back, instead of coming into the house, they sat down on the sidewalk a couple houses away and dumped each pillowcase into it's own separate pile. Then they traded out all of my chocolate for all of their Mary Janes and Bit o' Honeys. I had literally no chocolate or other preferred candy in my bag, it was all the crap candy no one ever wanted. I complained to my parents and was told that I should be grateful just to be getting any candy at all.
Two months later, both my sisters got chicken pox. My grandparents, saying how awful it must be to have chicken pox over the holidays, gave them extra Christmas presents. This is my lingering childhood resentment, and the injustice of it still stings.