The representatives of the third estate were chosen through elections. Breunig writes about them in his book:

"Delegates to the Estates-General had been named by voters who met in the chief town of each district. Members of the clergy and nobility had voted directly for their representatives, but voters from the Third Estate-That is, almost all male citizens over twenty-five-had named their delegates through an indirect system which had the effect of eliminating as candidates all but an educated elite drawn from the middle and upper bourgeoisie" (Breunig 1977, 8)

Palmer expands on it:

"For the Third Estate, in the towns, the lowest assemblies were meetings of the gilds and other occupational associations, to which another meeting was added for non-corporés who belonged to no such gild or association. Each of these various bodies deliberated, drafter a cahier, and sent deputies to an assembly of the town as a whole, which in turn deliberated, drafted a cahier and sent deputies to the principal district assembly, where they met with other deputies from other towns of the district and with the deputies sent by the peasants. The peasants, meanwhile, met in their villages, where all men twenty-five years old and listen on the tax rolls were admitted to the assembly, and where they too deliberated, drafted a cahier, and sent deputies to meet with those of the towns at the principal district assembly. Here, where townsmen and especially lawyers gained an easy ascendancy over deputies of the peasants, if only by greater fluency in public speaking and knowledge of public affairs, another cahier was drafted for the whole Third Estate of the district, and deputies were chosen to represent the Third Estate at Versailles." (Palmer 1959, 477)

The process varied by region, but essentially, people organized themselves in a corporate manner and sent deputies to the different town assemblies. Men over 25 that paid taxes and that were chosen by their town assemblies would participate in district assemblies which elected Third Estate representatives to the Estates General. In those assemblies, career bureaucrats and lawyers dominated because of their education. Furthermore, there was a rule that enabled maverick aristocrats or clergymen to represent the Third estate (eg Mirabeau, Sieyès): The First and the Second Estates could only be represented by one of their own. Anyone could represent the Third Estate provided that they were chosen by the latter.

But you are right, the estates general hadn't convened since 1614. There was no bureaucracy to support its election and no one had living memories of how it actually worked. Many went through the registers and the archives to figure out... what was the Estates General. Some even insisted that the process be the same as last time.

TLDR: Corporate groups=>Town Assemblies=>District Assemblies=>Estates General

Breunig, Charles. 1977. The Age of Revolution and Reaction, 1789-1850. New York: W. W. Norton & Company, Inc.

Palmer, Robert R. 1959. The age of the democratic revolution: a political history of Europe and America, 1760-1800, Vol. 1. Princeton: Princeton University Press.