7 vowel sounds (US pronunciation only): in 13 letters:
radioisotopic. The OED lists one US pronunciation as /ˌreɪdioʊˌaɪsəˈtɑpɪk/, with the 7 vowels /eɪ, i, oʊ, aɪ, ə, ɑ, ɪ/. The British pronunciation is only transcribed with 6 distinct vowels at most because the "i" in "radio-" is identified with /ɪ/.
7 vowel sounds (pretty solidly, I think) in 16 letters:
rhinolaryngology. The OED transcription is /ˌrʌɪnəʊˌlarᵻŋˈɡɒlədʒi/.
Since "/ᵻ/" is equivalent to "/ɪ/ or /ə/", this means this word can potentially can be pronounced with the 7 distinct vowel sounds
/ʌɪ, əʊ, a, ɪ, ɒ, ə, i/
This is a minor variation of one of the words mentioned in the following article: https://en.oxforddictionaries.com/explore/what-is-the-longest-english-word (otorhinolaryngological)
7 vowel sounds (but I don't think they're all legitimately distinct) in 12 letters:
radiobiology. The OED transcribes this as /ˌreɪdɪəʊbʌɪˈɒlədʒi/, giving 7 distinct vowels: /eɪ, ɪ, əʊ, ʌɪ, ɒ, ə, i/. However, I think few people actually distinguish the sound of the "i" in "radio-" from the sound of the "y" in "biology." That's why I don't think this is legitimate.
8 vowel sounds (but I don't think they're all legitimately distinct) in 26 letters:
radioimmunoelectrophoresis (another word from that article). The OED transcription is /ˌreɪdɪəʊˌɪmjᵿnəʊᵻˌlɛktrə(ʊ)fəˈriːsɪs/.
Why I think it's not legitimate:
As previously mentioned, the standard view, maintained by the OED, is "/ᵻ/" is just a notational shorthand for ""/ɪ/ or /ə/". So /eɪ, əʊ, (j)ᵿ, ᵻ, ɛ, ə, iː, ɪ/ does not constitute a set of 8 distinct vowels because /ᵻ/ is really the same as either /ɪ/ or /ə/. However, Janus Bahs Jacquet left a comment arguing that he does use three distinct vowels /ɪ/, /ə/ and /ᵻ/.
If we allow this, then I have another word with 7 vowel sounds in 15 letters:
radiomicrometer: the OED transcribes the parts of this word separately, the prefix "radio-" as /ˈreɪdɪəʊ/ and "micrometer" as /mʌɪˈkrɒmᵻtə/
The 7 vowels are:
/eɪ, ɪ, əʊ, ʌɪ, ɒ, ᵻ, ə/. I think this is a bit more legitimate since the first "/ɪ/" can actually be pronounced /i/, although the OED's transcription of British English does not acknowledge this.