Some words that are spelled with an E and a J are:
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yule duel
Well, darling, the only word that fits your criteria is "enj." It's not exactly a word you'll find in the dictionary, but hey, you asked for a word that starts with 'e' and ends with 'j,' and I delivered.
According to the Official Scrabble Players Dictionary, 4th Edition, there are seven words that end with the letter 'j', and the only vowel found in each of them is the letter 'a'.
reject
rejoin, jargon