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. The problem is almost certainly tied to the Play Store servers, so it's best to just wait until Google gets it sorted out.
The 403 errors appear to be hitting a relatively small percentage of users. There aren't any specific patterns emerging, but a lot of brand new and factory reset devices appear to have a higher than average failure rate. Perhaps they are being directed to newly provisioned servers that have been misconfigured.
Several people have offered different - fairly hacky - solutions that mostly center around clearing data from the Play Store and Google Play Services. It's possible that these methods will work, but they may cause other issues. Staring at a bunch of pending updates might be annoying, but it's best to wait it out, at least for now.