Hey Rebecca!
You can definitely record locally with Zoom (the main downside being you don’t get an auto-transcription that way). We’re recommending uploading to a private Youtube channel and setting the video to “unlisted” which will allow you to share the link with your students. Youtube has an auto-captioning feature which is helpful for accessibility purposes.
Warpwire is an option, but doesn’t have auto-captioning or auto-transcription.