How can JavaScript get data from a user's computer?

It sounds like you are flexible as to where the data is stored in the user's computer. In that case use localStorage or Web SQL Databases. Web SQL Databases are no longer part of the HTML5 spec, but they are supported by Chrome. Using either of these methods will store the data on the users computer - right in the browser in fact.

Edit: In the application that provides the caller id data, have it launch your website with chrome passing in a querystring argument containing the needed caller id data. Then in your page, have it listen to the onstorage event, and update the dropdown when the appropriate localStorage data has changed.

In your caller id app, call:

%ChromeInstallPath%\chrome.exe "http://foo.com/caller?Bill+Gates@425-882-8080"

In your caller handler, use this JavaScript:

window.onload = function() {
    localStorage.callerId = location.search;
};

In your page with the drop down list, use this JavaScript:

window.onstorage = function() {
    setDropDownFromCallerId(localStorage.callerId);
};