I found the following inscription on the cover of a "Codex vinorum": "Vinicultores Illustrissimi Hungari" (as I would translate: the most excellent Hungarian viticulturists). I would like to give this present to a Latinist and I have the possibility to have this inscription changed. Is it grammatically correct with the two attributives put after the noun? Is "Hungari" generally used in Latin language as an attributive or should "Hungarici" be used? Isn't it possible that instead of this inscription the following would be better: "Illustrissimi vinicultores Hungariae"? Thanks in advance!
The pair Hungarus (noun) and Hungaricus (adjective) in Latin is like "Swede" and "Swedish" or "Briton" and "British" in English. The corresponding pair for Hungary in English would be "Hungarian" and "Hungarian", but the distinction is clear in Latin. In this case you want an adjective, as you already have the noun vinicultor. In general, I would advice using the adjective demonym when an attribute is needed.
Latin prefers adjectives to genitives far more than English. For example, it would be well justified to translate Vinicultores Illustrissimi Hungarici as "the most excellent viticulturists of Hungary", but your translation is possible, too.
Putting two or more attributes next to a noun is indeed possible. Most adjectives follow the noun, but the ones expressing either size or a good or bad property typically precede it (cf. e.g. Tuomo Pekkanen, Ars Grammatica, §130.2). The adjective Hungaricus is neutral, but illustris describes something good. Latin word order is flexible, but there are some tendencies.
For these reasons I suggest Vinicultores Illustrissimi Hungarici or Illustrissimi Vinicultores Hungarici.