Kansas
Arkansas
Tennessee
Mississippi
Alabama
New Jersey *
Alaska

Interestingly, four of those states are neighbors.

I have gotten more email on this question than any other. They all say, "what about Maryland?!" and a few say, "New Jersey???" It's become necessary to add some discussion.

Why Maryland doesn't count

In elementary school we were supposed to learn that the vowels were 'a', 'e', 'i', 'o', 'u', and sometimes 'y' and 'w'. I never thought much about vowels again until I posted this question. Then it occurred to me to wonder, why "sometimes"; when does y serve as a vowel? The simple (and not exactly correct) answer is that 'y' steps in and serves then there are no other vowels in a syllable. Every syllable needs a vowel. For example, y is not a vowel in words like yellow. In a word like very, it is. Y serves as a vowel in Maryland's second syllable.

An extended email conversation with Larry Firrantello resulted in this statement, which, if it doesn't exactly answer the question, it does indicate that the question is much more complex than I'd ever intended it to be.

A letter from Larry, or, why New Jersey shouldn't count either...

I don't have a reference for you, but with four years of studying Linguistics I can offer you my own explanation.

What is a vowel and what is a consonant is a question Linguists could argue on and on about. There is more than one way to make that determination. You are using a functional definition, when you say that the Y in Maryland is a vowel. The vowel in a syllable carries the stress. However in English we combine vowels sometimes to make a dipthongs, sometimes using phonetic values like other languages do, but often we combine vowels simply as a way to write different vowel sounds. There are about 15 different vowel sounds in English and a, e, i, o and u simply do not suffice. The "long e" sound is written with different vowel combinations in "sea" "tree" "Willie" "receive" and the same way in "Jersey". Also the vowel sound in "mood" and "group" exists in "New". My point is that both Y and W are vowels in New Jersey. Note that W is only used in vowel combinations and never serves as the single vowel in a syllable (except "cwm" which is clearly considered a Welsh word) As if this isn't enough, arguments can be made that L and R can serve as vowels in English, and even M and N.