Since there was only one set of plates, we could never expect to find direct archaeological evidence of them, as they are now in angelic custody. However, there ought to be indirect evidence that suggests their likely existence, and that is all this question can seek.
The Book of Mormon points to a highly literate culture that existed in the Americas for around one thousand years. During this time, they must have written numerous books on a great many religious and secular topics. Even if none of the books survived, we should find fragments of writing material and writing implements, as well as graffiti on rocks or walls. There is no archaeological evidence that points to the possible existence of the gold plates that Joseph Smith assures us he saw in 1830.
The Book of Mormon is said to have been written in a reformed Egyptian characters, but no archaeological, linguistic, or other evidence of the use of Egyptian writing in ancient America has been discovered. in 1966, John A. Wilson, professor of Egyptology at the University of Chicago, wrote:
From time to time there are allegations that picture writing has been found in America… In no case has a professional Egyptologist been able to recognize these characters as Egyptian hieroglyphs. From our standpoint there is no such language as 'reformed Egyptian'.
The "Anthon Transcript" (also known as the "Caractors" document) is a small piece of paper on which Joseph Smith, Jr. is said to have transcribed reformed Egyptian characters from the golden plates. This could be a more direct evidence for the existence of the plates, even if not strictly archaeological evidence. Klaus Baer, an Egyptologist at the University of Chicago, called the characters of the "Caractors" document nothing but "doodlings."