He/might make/us a /vanillia cake. Am I correct to say 'He' is the subject, 'might make' the verb and 'vanillia cake' the object. What is 'us'and 'a'?

In addition to the above answer, the following gives a full syntactical analysis of the sentence. "He might make us a vanilla cake" follows the standard Subject Verb Object (SVO) syntactic structure for English sentences. "He" in this case in a Pronoun filling the Subject slot in the sentence. The phrase "might make" is the Verb "make" accompanied by the modal "might" indicating probability: "might make" is not definite. This modal verb phrase fills the Verb slot.

The Pronoun "us" answers the question "Whom" as in "He might make whom a vanilla cake?" "Us" is correctly identified as filling the Object slot as the Indirect Object of the Verb "might make." The rest of the Object slot is filled with the Direct Object ("a vanilla cake") of the Verb "might make": "He might make a vanilla cake." The words "a vanilla cake" form a Noun Phrase that fills the Direct Object portion of the Object slot.

The Noun Phrase "a vanilla cake" begins with a determiner followed by an adjective followed by a noun. A Noun Phrase is a construction that may include Determiners and Modifiers--which may be adverbs and adjectives ("all such artificially diligent workers")--and a Head Noun. In "a vanilla cake," "a" is correctly identified as a Determiner article; "vanilla" is a Modifier adjective, and "cake" is the Head Noun of the Noun Phrase unit "a vanilla cake."