How the quotes appear is determined by the font used. It's not Microsoft itself that controls the quote characters - it's the word processor's default font that you can - if you wish - select, although most start with the default of a fancy enough font that differentiates between start and end quotes. WikiTree's database (as most databases) does not use field content to determine display font face - there are other ways that the font face used on the web page is determined and WikiTree's edit page can only display the single character for start and end of the quotes, whether they are single or double. If you store the left or right version of a quote in the database then it cannot be interpreted when the page using that data item is rendered.
I find the easiest solution is, if I am composing something in a word processor and want to paste it into an edit field, I will first open a text editor (like Windows Notepad) and paste my content in there, then copy it from there to paste into the WikiTree edit box. The text editor will not choke on the curly quotes, but will automatically convert them to the straight quotes.
EDITED: Leigh Anne, the same key produces the start and end quotes, as you realize when you use the same key to type these things. It's the word processor's ability to interpret when a quote mark is starting and when it is ending the section that you're quoting - if the font face you're using includes what is called "smart quotes", then it will automatically convert the quote character you typed to either the start or end version. These use different ASCII codes, though, so when the WikiTree engine is producing the web page from the database content, it does not know what to do with those characters - therein lies the problem.