To complement @brianpck answer:
First, let's state right away the obvious: since all known species of llamas are endemic to South America, it is highly unlikely that there is a classical latin word to name them.
Wheeler (2005) offered a short review of the taxonomical history of llamas:
In 1758, Linnaeus described the two domestic New World camelids as Camelus glama “Camelus peruvianus Glama dictus” (llama) and Camelus pacos “Camelus peruvianus laniger Pacos dictus” (alpaca), placing them together in a single genus with the Old World dromedary and bactrian camels, Camelus dromedarius and Camelus bactrianus. The two remaining New World species, the wild guanaco and vicuña, were subsequently designated Camelus guanicoe by Müller in 1776 and Camelus vicugna by Molina in 1782. As early as 1775, Frisch proposed that the four New World species be placed in the genus Lama, but this work is not accepted by the International Commission on Zoological Nomenclature (Hemming, 1985a) and authorship of Lama is credited to Cuvier, 1800 (Hemming, 1985b).
But the nomenclatural history of llamas is even more complex than this as the Mammal Species of the World database page on the genus Lama Cuvier 1800 lists no less than 12 synonyms:
Aucheria F. Cuvier, 1830
Auchenia Illiger, 1811
Auchenias Wagner, 1843
Dromedarius Wagler, 1830
Guanaco Perry, 1811
Lacma Tiedemann, 1804
Lama Frisch, 1775
Llacma Illiger, 1815
Llama Gray, 1852
Neoauchenia Ameghino, 1891
Pacos Gray, 1872
Vicunia Rafinesque, 1815
To go back to the main point: it is not clear why Linnaeus used glama instead of lama or llama because, at the time, it was not compulsory (nor even habitual) to explain the etymology of a species name. Maybe Linnaeus just misheard the Quechua word, we'll never really know. However the reason why the genus name was changed to Lama could be because:
- Frisch and Cuvier wanted to correct Linnaeus mistake (if it is indeed a mistake)
- They wanted to avoid a tautonymous name: while Linnaeus seems to have been fond of them (being the author of Alces alces, Bison bison, Cricetus cricetus, Dama dama and many many others), the rest of the taxonomical community isn't. In fact the international code of nomenclature had to add an article specifically allowing them, to prevent the community from considering them invalids by default.
- Another explanation, maybe less elegant, is that the community erroneously consider Cuvier to have named the genus Lama: indeed the modern sources linking the name to Cuvier's work are pointing at table 1 from book 1 of Leçons d'anatomie comparée as the source from the name. But that table is actually written in french (and needless to say, Lama is french for llama), and indeed I do not seem to find any occurrence in Leçons d'anatomie comparée of the word Lama where it is employed in any way other than as a vernacular name in french. If it is true, at this point, it doesn't really matter anymore as the name has been employed for more than a century to designate that genus so usage prevails (particularly since Cuvier's work and the usage of the genus name Lama predates the first international code of zoological nomenclature).
If in the end the question was: what word to employ in a vernacular context (i. e. non-taxonomical) to designates llamas in latin? I would say that both glama and lama could be considered, though it might be worth noting that the latin version of wikipedia uses lama:
Lama glama (Linnaeus anno 1758), vulgo lama, est mammal familiae Camelidarum. Lama est indigena Americae Meridionalis et Andium montium, accurate Peruviae et Aequatoriae.
Source: https://la.wikipedia.org/wiki/Lama_glama. Emphasis mine.
Otherwise auchenia also makes sense to some extent (it derives from the greek auchēn meaning neck, and, as far as I know, it has no other meaning than llama).