[ol]
[li]bid[/li][li]bed[/li][li]bad[/li][li]bode[/li][li]bead[/li][li]bod (as in “has a hot bod”)[/li][li]bud[/li][li]bead[/li][li]boo’d (past tense of the verb “to boo”)[/li][li]bide (to "bide ones time)[/li][/ol]
That’s 10 words only distinguished by their vowel sounds; is that the maximum in English? Do other languages have more words that have the same consonants but different vowels (and thus different meanings)? Note: I’m not talking about spelling, just pronunciation). Have I missed any “b_d” words?
Homonyms don’t seem to meet the OP’s definition of “distinguished by their vowel sound” and “different vowels”. So you shouldn’t count both “bite” and “byte”, “bait” and “bate”, nor bad" and “bade” (unless you pronounce “bade” incorrectly).
The OP has “bead” listed twice, so there are only 9 words.
Rhotic and non-rhotic accents will complicate this, since some speakers will consider a word to have an “r” sound and others won’t.
Obviously we’re running into dialect issues here. I agree that baud and bawd are homophones, but in what dialect is baaed (what a sheep did yesterday) pronounced the same? You’re American, right?
Caught/cot merger, I presume. My American dialect is not part of it, but, roughly, about half of America is part of a dialect where “cot” and “caught” (and other words where “aw” vs “ah” are two different sounds) have the same vowel.
In addition to 9 from OP, and buoyed and bade from Riemann, we can get an even dozen by including
bowed - rhymes with loud
(Linguists who treat some 'r’s as part of vowel would have more — Beard, bird, etc. — but OP has banned these.)
OP is asking for words distinguished by vowel sounds, so if these are pronounced identically they don’t count. But I’m not sure they’re identical in all dialects. In fact, even ‘bad’ and ‘bad!’ may have different pronunciations in different contexts.