Update [2/12]: It looks like the glitch is over with. Several people are reporting that downloads are working again and everything has returned to normal.
If you've been having trouble with 403 errors while attempting to download new or updated versions of apps from the Play Store, welcome to the club. Reports have been popping up all over the Internet from people experiencing the same issue. Unlike the infamous Package File Invalid Error, the glitch appears to be persistent, preventing any and all downloads from starting.