Depending on what language you are programming in, you can use a function to replace double quotes with two double quotes.
For example in PHP that would be:
str_replace('"', '""', $string);
If you are trying to do that using SQL only, maybe REPLACE() is what you are looking for.
So your query would look something like this:
"UPDATE Table SET columnname = '" & REPLACE(@wstring, '"', '""') & "' where ... blah ... blah "