I would say: spell it any way you want, but be consistent. If your audience cares about the spelling, go with their choice.
As for whether or not you should directly quote what was said, despite your audience maybe not understanding, that's an interesting question. Probably you should write what they say, or at least ensure that you indicate how you are paraphrasing the quote.
If someone is passing away, then they are dying. Its origins are:
late 13c., "death," verbal noun from die (v.). From mid-15c. as a pp. adj., "in the process of becoming dead."
Its confusable "dieing" means:
To cut, form, or stamp with or as if with a die.
Finally, dyeing means to stain something with color, with the origins:
c.1400, verbal noun and pp. adj. from dye.
Why is dying the gerund/verb form of to die? For one thing, it follows a general rule of forming gerunds:
The vowel group -ie is changed to -y before adding -ing.
As this page also says this, it is safe to say that this is a generally accepted rule. Other similar verbs are tie and lie. It is not a special case, but I cannot find an exact date for when this rule was created. It is worth noting that, whether or not this is correct, there are early examples of the spelling dieing for dying in the 1800s. This leads to the thought that the currently accepted form is because of a spelling reform at some point in time since then.
Best Answer
The spelling vendor is the standard spelling. The New Yorker, as part of its bizarre house style, uses the spelling vender. No one else does, besides those trying to emulate The New Yorker’s style.
Of the 45 examples in COCA, only 17 were actual uses of the spelling vender outside of The New Yorker (compared with over 2000 examples of vendor, a ratio of over 100 to 1). Two were proper names, eleven were from The New Yorker, and fifteen were in foreign languages.