The video playback problem used to bother me a lot, it happens whenever an version update occurs, such as 7.0 to 7.1. Since the pre-installed chromium browser on raspberry pi os has no problem playing html5 video, so i think the existing shared object libraries might be workable on vivaldi. Here’s an easy way i tried to get things work without downloading files from snap:
1. update vivaldi browser to the latest version (currently 7.1.3570.58 Stable 64-bit )
2. copy libffmpeg.so from chromium to vivaldi directory and rename it as libffmpeg.so.7.1 to match the current vivaldi version
sudo cp /usr/lib/chromium-browser/libffmpeg.so /opt/vivaldi/libffmpeg.so.7.1
3. restart vivaldi
4. in order to check if proprietary media works, visit HTML5 Proprietary Media on Linux and the success video will play.

5. disable gpu-rasterization so the messy startpage and layout back to normal
vivaldi://flags/#enable-gpu-rasterization
It seems everything works fine after all
OS : RaspberryPi OS Bullseye Backport (arm64)
Vivaldi : 7.1.3570.58 stable (64-bit)
Chromium : 126.0.6478.164 Built on Debian , running on Debian 11 (64-bit)