This is actually an interesting question, but in order to answer it properly I'll have to provide a brief introduction of Hawaiian vocalism.
Let me start from the beginning. In Hawaiian, like in many other Polynesian languages, there are two kinds of vowels, short and long. Long vowels are designated with a macron (i.e. the horizontal diacritic symbol placed on top of a vowel), while short vowels are left unmarked.
Of course, two vowels can be combined together to form a diphthong. Hawaiian distinguishes between short diphthongs (i.e. short vowel + short vowel) and long diphthongs (i.e. long vowel + short vowel). In principle, these are the only sequences that you would see most of the time in most phonetic descriptions of Hawaiian (ʻŌiwi 2018: 107-110). We can claim that Hawaiian has eight short diphthongs, i.e. au, ai, ao, ae, eu, ei, ou, oi, and six long diphthongs, i.e. āu, āi, āo, āe, ēi, ōu. Hawaiian diphthongs have a distinctive prosodic feature, they are all stressed on the first element of the sequence.
Vowel sequences (like io, iō, iu, etc.) other than those I have mentioned above are simply vowel clusters. These differ from diphthongs in that their second element is the one bearing the stress (Elbert & Pukui 1979: 15). Moreover, both vowels that form a vowel cluster can be either short or long.
In light of these considerations, while āo is indeed a diphthong, the sequences āō and aō are vowel clusters. Now, the reason why you don't see āō or aō quite often in dictionaries (especially modern ones) is that there are very few words that contain them. In Elbert & Pukui's (1979: 15-16) grammar, you can find the verb kāō "to go together in a crowd" and the place name Keōkea "The White Sand". The latter shows the presence of the vowel cluster eō, which has a long second element, like aō.
But why are these vowel clusters so rare? Sequences having a short vowel plus a long vowel, or two long vowels, arise in Hawaiian due to (a) the addition of a suffix starting with a long vowel to a stem ending in a short or long vowel, (b) borrowing (like Kāpāōka < Eng. Sabaoth), or (c) a peculiar phonetic process of syllable elision that I will now describe in more detail. In very few compounds and words derived through suffixation, a stem that is at least disyllabic looses its last syllable and lengthens the preceding vowel. Elbert & Pukui (1979: 16) provide some examples of such phenomenon: the words moku "district" and kapu "sacred" make up the place name Mō-kapu "The Sacred District", where the original disyllable moku is reduced to mō-; the word one "sand" becomes -ō- in the place name Ke-ō-kea "The White Sand"; the words maka "eye" and kahi "one, single" give rise to mā-kahi "one-eyed, monocle"; and finally limu "seaweed" is reduced to lī- in the compound lī-pahapaha "a kind of seaweed". Therefore, vowel clusters with a long second element have a secondary origin, and some of them can be considered outputs of this phonetic rule whenever a syllable was dropped after a diphthong.
This explains why vowel sequences like āō and aō are so rare, and are often not considered as regular vowel sequences in most grammars and dictionaries. In conclusion, such sequences do exist: they are not diphthongs however, they are vowel cluster, and are almost exclusively found in compounds and loanwords. I hope this answers your question.
- Elbert, Samuel E. & Pukui, Mary K. 1979. Hawaiian Grammar. Honolulu: University of Hawaii Press.
- Elbert, Samuel E. & Pukui, Mary K. 1986. Hawaiian Dictionary. Hawaiian-English, English-Hawaiian. Revised and Enlarged Edition. Honolulu: University of Hawaii Press.
- ʻŌiwi, Parker J. 2018. Hawaiian. Journal of the International Phonetic Association 48, 1: 103-115.