What do you call someone who studies letters?

You could use epistilologist or epistilographer. However these words tend to mean both those who study letters and those who are skilled in the art of letter writing.

"Epistilologist" is a description associated with the Russian historian V.A. Smetanin, who specialised in the study of letters (see Hatlie). "Epistilology"/"Epistilologist" are not in the OED or other dictionaries but are found in academic papers. But sometimes "epistilologist" has the sense of an expert letter writer (e.g. Georg Simmel), rather than one who studies letters.

"Epistilography" is a term for the study of the art of letter writing, particularly related to those of the Byzantine empire, but not exclusively so; it appears in some dictionaries. Words ending in "graphy" can form nouns for experts or practitioners with "-er" or "-ist", but neither epistilographer nor epistilographist seems very common and don't seem to be in most dictionaries. (The OED includes epistolographer as meaning a writer of letters, although it seems to refer to a habitual or expert writer, one practiced in the art of letter-writing, e.g. from 1824 "Marcus Tullius Cicero, at once an orator, a philosopher and epistolographer.")

Note that letters are studied in many disciplines: communication theorists or communication scientists study how to facilitate communication and problems in communication; historians study letters as historical evidence; academics studying literature (literary theorists or literary scholars) use them either as literary texts or as evidence to understand past writers; theologians may study Biblical letters; other disciplines such as art history may also make significant use of them. The above terms refer to historical, literary, and theological study, but you may want to be more specific.

References:

• Hatlie, P. (1996). Redeeming Byzantine Epistolography. Byzantine and Modern Greek Studies, 20, 213-248. doi:10.1179/byz.1996.20.1.213
• The Routledge International Handbook of Simmel Studies, edited by Gregor Fitzi,
• Merriam-Webster, "Epistilography"
• "epistolographer, n.". OED Online. March 2021. Oxford University Press. (accessed March 24, 2021).