I'll be livestreaming a funeral in a couple of weeks for the family.
Are you running the livestream, or is someone else and you want to capture it?
If the event is being livestreamed by someone else, ask whoever is doing so if they'd record it. And if Facebook or YouTube (or similar), the recording is available for future viewing as is (and there are ways to download those videos without going through effort to capture/re-encode, with resulting drop in quality).
When streaming, I also record locally at 1080p as the stream to FB is at their required 720p. For a Baptism, I then uploaded the locally recorded (higher resolution) version of the stream for the family (and will do the same for the funeral, if requested). Oh, and my streaming PC also has a DVD burner so I can provide that content to those that are Internet challenged... s I'd recommend just asking whoever is doing the service what options they have