Can I store my backups with UpdraftPlus?

Yes. UpdraftPlus has various options for remote storage – e.g. Dropbox, Amazon S3, Google Drive, FTP, etc. Naturally, it has to actually create the zip file(s) to be stored remotely before it can then send them to the remote storage and then delete them from your site after sending them. There will be a short time gap in which the backup exists on your site before it is dispatched and deleted.

This should be fine with all web hosts that say that they allow you to create backups and store them remotely – because there’s no other way of doing it. It should be especially fine if you use UpdraftPlus’s Premium scheduling feature runs the backup in the early hours when the web server will only be lightly loaded. If they don’t like it being done that way, then they are basically forbidding you to make a zip-file-based backup. Time to find a web hosting company that’s not selling crippled hosting.