Vexed means annoyed or frustrated, so you could substitute one of those words in the sentence to help you hear which preposition sounds best. With is the first one that springs to mind, but others would be 

• about
• concerning
• because of
• by

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. 

In my opinion, there are two prepositions that could be used to complete this sentence.  You could use "with" or you could use "at."  So the sentences would be

• He is vexed with me.
• He is vexed at me.

Of these, I would prefer the first one.  In American English, at least, it sounds much more natural.  However, we do not really use "vexed" very much in normal conversation so it does not sound completely natural either.

I do not think that any other prepositions would fit in this sentence.