Just wanted to thank everyone here for leading me to a solution to getting a video to play locally. What I did was create a simple html file with only the following:
<video id="video" width="1280" controls>
<source src="video.mp4" onclick="this.play();">
<track src="testCaptions.vtt" kind="subtitles" srclang="en" label="English">
Then, in an Axure Inline Frame I checked "Link to an external url or file" under Frame Target, and typed only the name of the html file, something like "dummyVideoFile.html".
When you publish to html..., just add the "dummyVideoFile.html" and "video.mp4" to the root directory (or in a folder, but then you have to point to that folder in the html file). This worked perfectly, and used the browser default video player right in my project.
Now if only Chrome would let us load in local .vtt caption files like Firefox, THAT would really make my project complete!