How should I denote drawn-out vowels in English? If I have a character with speech disorder or with a very unique accent, what is the correct way to express, in written form, this quirk of their speech? Would it be appropriate to just repeat the vowel? Or perhaps there's a specific punctuation mark I am not aware of.
I think you need to explain it in your narrative in most cases, unless it is something that can be approximated with additional letters (such as stuttering or lisping). You can also use the way it is pronounced in a dictionary or elsewhere, such as, "meez-on-senn" for mise-en-scene. If you haven't read it for awhile, you might want to re-read a couple of chapters of the great American author Mark Twain's Huckleberry Finn, to see how one novelist managed to do it and remain both credible and engaging without stumbling over his own words or those of his characters.
Grammar Girl (among others) recommends avoiding it:
It may say more about the author and his or her assumptions than about the characters, or it may distract readers to the point that what is being said is overshadowed by how it's being said.
The article goes on to make this recommendation:
The other option for communicating a character's accent to readers, which I recommend, is to use standard spelling along with a description of the character's speech in the text introducing the character. One might write, "Her roots in the South were evident in her slow, melodious speech," while using standard spelling when writing out her speech. This method is much easier for the reader and avoids inadvertently stigmatizing a character.
Yes, you would repeat the vowel. Be careful not to double a vowel that would form another word (e.g. elongating the vowel in to to form too actually makes another, different word). For this reason, writers will usually use at least three consecutive vowels, such as:
I was sooo happy to see them.
That was waaaaaay too much effort.
I reeeeeally can't believe it.
There is no official rule about how many iterations of the vowel to use, other than the rule of thumb that more = longer when spoken.
There is also no official rule about how to handle the second vowel when two letters are used together to form a single vowel sound (as in really in the example above). My preference is to elongate the letter that most closely approximates the sound intended, as in:
Sometimes you will also see the second letter doubled, perhaps to a lesser extent (e.g. booooooaaat) but this is slightly harder to interpret (in my opinion).
Understand, of course, that the literal text of what someone says, does not change when spoken with a heavy accent or mispronunciation. So, if you are quoting someone in a journalistic sense, you should always write their words exactly as the speaker intended, and with the correct spelling.
To do otherwise, is injecting your own opinion, making a caricature of the person, by exaggerating some aspect of how they happen to speak. However, if you're writing a fictional story, or even a non-fictional accounting of a real-life "character", you may wish to illustrate that individual's characteristic style of speaking.
Because there are so many possible accents or ways to mispronounce words, there are no official rules of how to achieve this — it's up to the author to decide how the spoken words sound, and how to write them down phonetically.
Depending on your own regional accent, certain variations that other people have may stand out to you more than others. If a beautiful picture is in the eye of the beholder, the perfect accent is in the ear of the listener. So use your ears, and write down what you hear.
To give you a real-life example, I live in an area of Massachusetts known for its strong regional accent. The mayor we had a few years ago, spoke it so well, that I saved a few voice mail recordings of alert messages he sent. Here are two that you can listen to:
There are many words in the mayor's recordings that exemplify his authentic local accent. Notably, when the mayor says:
" There is no parking on emergency arteries. Cars parked on such arteries will be ticked and towed. "
" This roadwork is a necessary part of the construction of the Wonderland parking garage. "
If I were to write that in a way that illustrates how it sounds to me, it would be:
" There is no paahking on emergency aahtahries. Caahs paahked on such aahtahries will be ticked and towed. "
" this roadwork is a necessary paaht of the construction of the Wonderland paahking gaahradge "
So, it's really a subjective thing, but it should only be done when you're more concerned with expressing the sound of someone's speech, rather than the meaning of their spoken words.