Learn German: Wir kaufen das Schränkchen und das Stühlchen. - We are buying the little cabinet and the little chair.

You are right that the original nouns are der Schrank and der Stuhl. However, whenever you add the diminutive suffix -chen (or -lein) to make something little, the new word automatically becomes neuter. That is why they must change to das Schränkchen and das Stühlchen.

Why do the vowels change to an umlaut in Schränkchen and Stühlchen?

When creating a diminutive with -chen or -lein, the main vowel of the root noun usually takes an umlaut if it is an a, o, or u. Therefore, the a in Schrank becomes an ä, and the u in Stuhl becomes an ü.

Since we are buying them, shouldn't the nouns be in the accusative case?

They are! The verb kaufen requires the accusative case for the direct objects being bought. But because both items are neuter nouns, their definite article das looks exactly the same in the accusative case as it does in the nominative case.

How would this sentence look if we were buying multiple little cabinets and little chairs?