Can I access a property from my manifest.json in my extension's JavaScript files?

Since chrome 22 you should use chrome.runtime.getManifest(). See docs here.

So now it is as simple as:

var manifest = chrome.runtime.getManifest();
console.log(manifest.name);
console.log(manifest.version);

I think that this is what you're looking for http://www.martinsikora.com/accessing-manifest-json-in-a-google-chrome-extension

chrome.manifest = (function() {
    var manifestObject = false;
    var xhr = new XMLHttpRequest();

    xhr.onreadystatechange = function() {
        if (xhr.readyState == 4) {
            manifestObject = JSON.parse(xhr.responseText);
        }
    };
    xhr.open("GET", chrome.extension.getURL('/manifest.json'), false);

    try {
        xhr.send();
    } catch(e) {
        console.log('Couldn\'t load manifest.json');
    }

    return manifestObject;

})();

And that's all. This short code snippet loads manifest object and put's it among other chrome.* APIs. So, now you can get any information you want:

// current version
chrome.manifest.version

// default locale
chrome.manifest.default_locale