No. That is quite frankly not possible.
There are, however, Japanese names like Ai that only have vowels
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The words "I" and "a" each contain only one letter which is a vowel. a word that has only vowels could be aiea and it is a city name
The only one of the United States with three consecutive vowels in its name is Hawaii
Hawaii.
all words have vowels
The only related term I have been able to find is: consonantal - of, pertaining or including consonsants. (This definition does NOT however specifically exclude the use of vowels). To do so, I'd suggest adding "-only", i.e., consonantal-only words. For me, it's simpler to say "consonant-only words". BTW, The antonym for consonatal is vocalic (of/pertaining to vowels).