Could the plot of Rebecca make sense if the setting was America in the 21st century? Why or why not?

The plot of Rebecca could make sense if it were set in twenty-first century America.
Two elements would have to be in place: first, a similar lack of information about who Rebecca really was and second, a narrator so insecure and lacking in self-worth that she would readily assume that Rebecca must be infinitely superior to her.
Admittedly, both elements would be more difficult in twenty-first century America, which is the implication behind the question. Stories don't always just transfer seamlessly from one culture to another.
In our culture, the differences are that we have the internet and, thus, far more easy access to immense amounts of information than the second wife. Further, we live in a culture than brings up both males and females to be assertive and have a high sense of self-worth.
Nevertheless, these are not insurmountable—or even, in the end, difficult—obstacles. Americans today, and almost everyone else, have the internet and social media at their fingertips, but fake news and disinformation are also rampant. People even hire PR firms to make sure search engines show only tops hits about them that are positive. They control their image and carefully craft a fake self that appears on social media. One can easily imagine Rebecca doing that (of course, she would), and Mrs. Danvers maintaining the carefully idealized legacy.
Second, while we raise children to be more assertive, as we know, levels of suicide have gone up in recent years, and many young people have a low sense of self esteem. We can easily imagine creating a narrator who has been bullied at school, who grew up in an unstable home, and who cut herself or attempted suicide in high school. Such a person would come into this situation without having sufficient ego or confidence to challenge the version of the Rebecca story she is told.

The plot of Daphne du Maurier's Gothic classic Rebecca hinges on the fact that the unnamed narrator does not have and cannot easily get all the information about her new husband's ex-wife, the beautiful Rebecca. The narrator's husband, Maxim de Winter, refuses to talk about Rebecca, so the only sources the narrator has for information on Rebecca are her portrait and her loyal and therefore unreliable housekeeper, who is still in charge of maintaining Maxim and her house.
I could see arguing this question either way. A Rebecca updated for 2018 could make excellent use of social media to feed the heroine's obsession. Now, instead of staring at her portrait, the narrator would scroll through the dead woman's instagram feed. Her jealousy would be fueled not only by what the fawning housekeeper says about her but also by all the fawning messages of grief people who barely knew her have now posted on her facebook wall.
That said, because of social media and the internet at large, the narrator would have much greater access to information about Rebecca, and it's possible that through this access she would figure out on her own that Rebecca was, in actuality, no one to be jealous of. There have also been many advances in detective and police work, and it's possible that with things like DNA evidence at play, the late Mrs. de Winter's death would not go unsolved for as long as necessary for the events of the plot to unfold.