Both spouses must be present to marry in almost all U.S. jurisdictions.
A handful of U.S. jurisdictions that permit people to marry when they are not both physically present at the same place. But, even in those states, this is allowed only in very limited circumstances (mostly foreign military service deployments of one spouse). Also, even in those states, there must still be a proxy for the missing spouse at the marriage event to approve the marriage, who has been expressly authorized by the absent spouse to serve in this role. This is called "marriage by proxy."
Other common law legal jurisdictions mirror the U.S. in this regard.
In almost all civil law jurisdictions, the essential point is that both spouses sign the marriage form (not necessarily contemporaneously in some jurisdictions), and the religious marriage ceremony has no legal effect.
The only circumstance I can think of which is an exception to this rule is that France allows a person who was formally engaged to another person, but died before the marriage could be concluded, to be married posthumously in some circumstances.
The presence of both spouses at a wedding has been required at all times in human history of which I am aware, even in places outside both the common law legal tradition and the civil law legal tradition, such as traditional tribal law, and such as the laws of Asian, African, and American civilizations prior to meaningful Western influence on their legal systems.
The assent of both spouses has been required almost everywhere for hundreds of years almost everywhere (although in pre-modern times, there was less of a concern that party to the wedding might be doing so under duress), and there have been some times in the historical record when a woman's consent to her marriage was not required and a parent or guardian of the woman gave consent to her marriage instead. But even then, the woman's presence at the wedding was required.
Specifically:
Two people, of the opposite gender, are planning on getting married. When it is time for the ceremony, one of them does not show up. The minister that is performing the ceremony marries the two people anyway.
Assuming the person that did not show up did not want to get married, what should that person do? I am thinking he/she can and should apply for a legal annulment. Am I right about that?
The marriage would be void ab initio without any action taken, but if a legal document such as a marriage certificate were filed anyway, purporting to show that the couple was married, a legal action for an annulment would be an appropriate action. Likewise, in the Roman Catholic church, at least, an annulment would be a remedy available for religious purposes under canon law and the priest who conducted the wedding anyway would also be severely punished for going forward with the wedding in this situation.
I also want to know, if this has ever happened in real life. I believe it has ; at least in the past.
I am not aware of a single example of this happening ever, in the recorded history of the world. (Who knows what happened before then, when law as we know it wasn't really a thing.) But, recorded history is about 5500 years long, and involves tens of billions of people all over the globe, and I'm not omniscient.