Ha! Download the list of English words, then
Not all of these words are in Merriam-Webster Dictionary, though. Some are in Oxford English Dictionary or in Wiktionary. In fact, Wikipedia says this.
$ grep '^[^aeiou]*a[^aeiou]*e[^aeiou]*i[^aeiou]*o[^aeiou]*u[^aeiou]*$' words.txt
One does not mess with regular expressions.
There are many words that feature all five regular vowels occurring only once in alphabetical order, the most common being abstemious and facetious. Two of the shortest, at eight letters, are caesious and anemious (OED); and aerious (OED) has only seven letters. Some others are abstentious, acheilous, arsenious, arterious, tragedious, fracedinous, and Gadsprecious (all in OED). Considering y as a vowel, the suffix -ly can be added to a number of these words; thus the shortest word containing six unique vowels in alphabetical order is aeriously, with nine letters (OED); the much more common facetiously has eleven letters.