I believe the answer is,
4085 (four thousand and eighty-five)
962000 (nine hundred and sixty-two thousand)
First I made a table of all words and vowel appearences.
... full table can be found at https://paste.ubuntu.com/p/X8sfQnhVFK/
From the table it's pretty clear no number <100 could work. We show why numbers in 100~999 couldn't work.
Those numbers have format:
 hundred (and -/) (here  stands for a word)
We need 7 vowels, however one word (except the -teen case, where 2 words can be used then) contains no more than 2 vowels and we only have 3 slots.
Now 4 digit numbers. Plain
 thousand certainly won't work. Notice that
and both contain
a, so we can at most have one
Two cases left:
 thousand and  hundred and
 thousand and -/.
Looking through all digits, sadly there aren't two digits matching this need.
Second case looks more promising.
As before, we need two or three words here with 6 vowels in total. Our only bet is to have three words, all with 2 vowels.
The only -ty's with 2 vowels:
If we choose
seventy, we're left with
00211 and sadly no match again. If we choose
eighty instead, we're left with
five is such a match.
6 digits, so a few hundred thousands.
 hundred thousand won't work. As said, we can only have at most one
and pairing with
thousand. Three cases:
 hundred and  thousand,
 hundred thousand and  hundred,
 hundred and -/ thousand (There's also
 hundred thousand and -/ but is certainly smaller).
The second case has 3
us so not valid and the first case is identical to the
 thousand and  hundred discussed in the first question. The only bet is the third case.
Two digits won't work. -teen's have two
es. So we're going for the three word case.
To make the number as big as possible, we put the largest digit
nine in the front.
Two vowels left. Again we choose the biggest possible number to put with one vowel, which is
sixty. We're left with the one last
o so the last digit is
On the bonus question,
The answer is yes. For example, 3000000000084000 (three quadrillion and eighty-four thousand). I'm not sure if it's the smallest one (hopefully not).