iCloud web UI. Sorry, I just tried again and you can add tasks, but you can't set a due date or time, which I rely on heavily. You used to be able to do this until they went to the "new" Reminders. They gave us a prompt to upgrade to the new reminders, but you don't realize that you lose these features, and as far as I can tell, you can't go back after you "upgrade".
I just set a due date and time on a reminded with the web UI. Add the item, click "details", click "on a day", set a date and time. It's not easy, or a nice UI, but it's functional for this visual user. I didn't check the accessibility.
I just used iCloud.com. Seems to work fine for adding reminders. Are you talking about an API, or the iCloud web UI?