I am trying to research how new species have been recorded over time, but specifically in the period before the 1930s. Where would an academic who had discovered a new species get their discovery ratified and recorded for posterity over that period? Were there multiple registries in different countries? How centralized was the process? I am particularly interested in marine species discovered in the USA as I may have had a relative involved at the time.
Today, to describe a new species, you need to publish an article containing a description of the new species in a scientific journal that adhere to the rules of the International Code of Nomenclature relevant to your species (as there is a different one for animals, plants, bacteria, etc.), you need to name it according to the rules (and if possible recommandations) of said Code of Nomenclature, and most importantly, you need to deposit the name-bearing specimen (a. k. a. the holotype) in a collection (most often in a museum but not necessarily, as long as the collection is correctly curated, and accessible to any other scientists, on request) where it is given an accession number.
But, the 1st edition of International Code of Zoological Nomenclature (in your case, I suppose) was only published in 1961.
Species described before that date, often had an holotype (as the concept was older than the ICZN) but not necessarily: all that was necessary was the publication process, with a clear illustration if possible or an unambiguous description (note that pre-1900 species descriptions rarely are but the name can be conserved if it is commonly used by the scientific community). Often, if necessary, a holotype can be designated a posteriori either from a series of specimen known to have been used by the original author to describe the species (in which case the new name-bearing specimen is a lectotype) or a brand new specimen if none of the original specimens have been conserved (in which case it is a neotype). Neotypes are designated only when deemed absolutely necessary, as they are by nature more dubious than lectotypes.
What could make the species name unavailable is if the name was not published in accordance to Article 8 of ICZN, if the name use another alphabet that the latin alphabet (diacritics, hyphen, apostrophe are not a problem though, but should be corrected a posteriori), if the work that contains the new name didn't make a consistent use of binomial nomenclature, if the name was already considered a synonym at time of publication or if the species name was not published unambiguously as being combined with a genus name.
Specifically, species names published before 1931 are subjects to the previous rules of name availability (known as Art. 11) and to this requirement (Art. 12): "To be available, every new name published before 1931 must satisfy the provisions of Article 11 and must be accompanied by a description or a definition of the taxon that it denotes, or by an indication.".
To answer one of your secondary questions: the process was not and still is not centralized. There is no global registry of new species names (there is Zoobank, though, which is a registry that is mandatory but only for species published in online-only journals, see Art. 8.5.3 of the ICZN).