In a way, vowels are already biphonic! Acoustically, vowels (and most sounds, actually) are simply combinations of formants: specific frequencies at which the vocal tract resonates. The differences between vowels are then caused by differences in the frequencies of the formants. This can be easily seen on a spectrogram, like this one from Wikipedia:
In this spectrogram, the vowel [i] has its first two formants at roughly 500 and 2500 Hz, while [u] has its first two formants at roughly 500 and 1000 Hz.
Now, it turns out that biphonic singing works on exactly the same principle! Biphonic singing works through shaping the mouth so that one of the formants becomes loud enough to be perceived as a separate note. Note that this uses exactly the same mechanism used in vocalising vowels; the only difference is that the shape of the mouth is changed slightly to emphasise the formants. To experience this yourself, you can try saying [u͡ʉ͡y͡ʉ͡u] very slowly; if you listen carefully to what you are saying, you should be able to hear a note growing higher and then lower in pitch. This note is one of the formants of the vowels (the first formant, I think); Biphonic singers simply emphasise this formant so it becomes loud enough to easily hear.
So, to summarise, languages already use vowels which could be described as ‘biphonic’ — the only reason we don’t hear it that way is because the formants aren’t as strong as they are in biphonic singing.
(For more information, you may find https://newt.phys.unsw.edu.au/jw/voice.html and https://newt.phys.unsw.edu.au/jw/xoomi.html interesting.)