Yes, you can use a semicolon in front of and. But to do so you would need to rephrase the sentence:
There are two things required of pupils: they must know what adjectives, nouns, and synonyms are; and they must be able to infer the meanings of adjectives based on context clues.
Here, a colon is used to to introduce an explicit list of items. In such constructions, a semicolon is a common stylistic feature used to clearly delineate each list item from the rest when one or more list items already contain commas and the use of a comma as a delineator would be confusing.
But while technically correct, it looks odd. Normally, a semicolon would be used in a construction with three or more list items, not just two. There is nothing that makes it wrong but it's nonstandard and I've never actually seen it done. (Perhaps it's wrong from the perspective of common style.)
Your sentence could maintain the distinction between the two things and look less strange if you replaced the final and and turned it into two independent clauses:
Pupils must know what adjectives, nouns and synonyms are; they must also be able to infer meanings of adjectives based on context clues.
I should also add that, per a comment, there is nothing wrong with this construction:
Pupils must know what adjectives, nouns and synonyms are and be able to infer meanings of adjectives based on context clues.
The only reason to rephrase it is if you don't like it (you can write things as you wish) or you think other people might find it confusing. But in terms of grammar, there is no reason why it couldn't be used.