Prepositions are words which link nouns, pronouns, and/or phrases to other words in a sentence. Prepositions serve the purpose of clarifying the relationships between particular words; they are generally directional (direction prepositions such as "up" and "down"), positional (place prepositions such as "against" and "around"), or indicative of time (time prepositions such as "after" and "before"). Prepositions occur within a sentence in front of a noun.
So, as the other educators have pointed out, there are two options here for filling in the prepositional blank:
- He is vexed AT me.
- He is vexed WITH me.
Which you use is a matter of personal preference. I find that it is most common to hear "with" used within this context, and I personally find it more pleasing to the ear.