For the past few years we've recommended the ScreenCastify system for making quick recordings of web sites for both tutorials and for sharing with us problems that you are seeing with your web site. Unfortunately, ScreenCastify is a Chrome-only browser plugin. They presently don't appear to have a Firefox plug-in even in development.
If you haven't found something else, we are now using and recommending Loom.com.
There is a free plan that works just fine for most (esp. if just reporting web site issues), and their Pro and Business plans provide all the features that most would need.
We're still getting started using it but if you decide to use it and have questions feel free to ask and we'll try to help.