John L. Sorenson discusses "synagogues" in the Book of Mormon; argues they existed in preexilic times and there were structures in Mesoamerica that could be labeled "synagogue."
John L. Sorenson, An Ancient American Setting for the Book of Mormon (Salt Lake City and Provo: Deseret Book and the Foundation for Ancient Research and Mormon Studies, 1985), 235-36
What were synagogues? They are mentioned among both Nephites and the Lamanites under dissident Nephite influence (Alma 21:4-5; 32:1-12; Helaman 3:9, 14; Moroni 7:1). Would they have left ruins that might have been discovered? At first glance the very idea seems to pose a problem for the Book of Mormon. Many historians have maintained that synagogues were not known among the Jews until well after Lehi had left Palestine. Another group of experts, however, now argue that the synagogue predated Lehi's departure. They propose that when King Josiah carried out his sweeping reforms of Jewish worship in order to clean out pagan intrusions, he closed the old sanctuaries (2 Kings 23). "The centralization of worship in Jerusalem from 621 B.C. onwards, with many Jews thereby denied a share in temple worship, must inevitably have led to the establishment of non-sacrificial places of assembly"—in effect, synagogues. So at least the concept of the synagogue could well have been around for a generation by the time First Nephi begins. Later synagogues served as community centers open to any who wished to worship or speak (compare Alma 26:29). According to the Babylonian Talmud, the Jewish synagogue was normally oriented to face Jerusalem and was also located on the highest place in town and near water. A synagogue was not necessarily a building; it might be only an enclosure.
Structures for seemingly sacred purposes that meet most of the Talmudic criteria existed in early Mesoamerican sites. It remains for some ambitious student to make detailed comparisons. That study should look carefully at names as well as ruins. The term synagogue is difficult to distinguish in concept from related terms used in the Book of Mormon. The "churches" set up by Alma in Zarahemla, and also the "assembly" of the Lamanites (Alma 21:16), were apparently functional parallels to synagogues. Several Old Testament terms signify "congregation" or "assembly" or the meeting place for such a group, the terms overlapping in translation. One of those words has come to be translated "synagogue," but anciently words like synagogue, ekklesia, kenishta, and 'eda were translated quite freely as though they were equivalent. Thus, we may find that whatever distinguished a synagogue from a local church by Nephite standards was so subtle that we will be unable to tell them apart on the basis of their remains.