How many vowels are there in the English language?

The name "vowel" is often used for the symbols used for representing vowel sounds in a language's writing system, particularly if the language uses an alphabet. In the Latin alphabet, the vowel letters are A, E, I, O, U, and Y.
There is not necessarily a direct one-to-one correspondence between the vowel sounds of a language and the vowel letters. Many languages that use a form of the Latin alphabet have more vowel sounds than can be represented by the standard set of five vowel letters. In the case of English, the five primary vowel letters can represent a variety of vowel sounds.

https://en.wikipedia.org/wiki/Vowel#Written_vowels

I agree with gdec with the proviso that Y is both a consonant, as in yet, and a vowel, as in my. At one time, J performed a dual function as well, as did V. However, you will also find W as a vowel in loanwords like cwm, borrowed from Welsh. Yes, that is now a word in English. It's as English a word as garage, shampoo, kiosk and jaguar.

If this is a question about English speech, not written English, there are many vowel sounds.
Have a look -
https://www.babbel.com/en/magazine/english-vowel-sounds