The short answer is that oriental is the adjective associated to the noun est. “Eastern X” and “X oriental(e)” are in straightforward correspondance.
In French, you can't just take a noun and use it as an adjective. You often need to add or change a suffix, and sometimes more than that. There is no adjective built directly on est¹.
It's very common for “basic” nouns and verbs to have evolved continuously from Latin (formation populaire), and for more “fancy” terms to have been introduced during the Middle-Ages in a form that's a lot closer to the Latin (formation savante). There's a somewhat similar phenomenon in English with basic words of Germanic origin and fancy words of Romance origin. The cardinal directions in French (est, ouest, nord, sud) actually come from English (or more precisely Anglo-Saxon), and can be considered formations populaires. They have associated adjectives that come from the names of the cardinal directions in Latin: oriental (from oriens), occidental (from occidens), septentrional (from Septentrio), méridional (from meridies). The adjectives oriental and occidental are the “normal” adjectives for east and west. Méridional is a little less used, with sud being more common as an adjective or apposed noun¹. Septentrional is rarely used and there is an adjective nordique, but this adjective is mostly used only with the same meaning as Nordic in English, referring specifically to the Nordic countries.
Other constructions would be possible, for example “Suisse de l'est” (like Corse-du-Sud, Corée du Sud) or “Est-Suisse” (like Nord-Kivu). But “Suisse orientale” follows a common and (to a French speaker) unremarakable pattern, like Pyrénées-Orientales, Flandre-Orientale and Flandre-Occidentale,
¹ The word est is sometimes classified as an adjective in expressions like “la rive est” (the east coast/bank) and “le versant est” (the east flank [of a mountain]), but I think it would be more accurate to classify it as a noun in apposition, since it is invariable. Adjectives normally agree with the noun in French.