That's true. Bastillion mentions its available in the aws marketplace, so it could be an option to run it in aws.
I was thinking there might be web terminals available, e.g. https://www.serfish.com/console/ . I don't know anything about them though, and you'd want to trust the provider.
I've got AWS but I can't find any web-based ssh that doesn't require a download. There are some other links in this post that I haven't checked out yet but I did do the initial research.
What do you mean by web-based shell? Would that be different from a web-based ssh client?