No at this time Vidyard does not push captions to Facebook with the Facebook push function, but you can add them yourself after downloading them from Vidyard.
Download the caption from Vidyard. Upload to Facebook.
Navigate to the video in Vidyard.
In the list of captions, select the caption that needs to be added. Download the .srt version.
Adding Closed Captions to a Facebook Video. After the Upload is completed from Vidyard, click on your video to expand it. Click Options and select Edit This Video from the menu. Click Choose File to select your SubRip (.srt) file. Click Save.