Am I allowed to teach my online students someone 's book?
Copyright is about copying, not about use. So, it is possible to do this if you use a bit of caution. First, though, you should ask the publisher about fair use in this case. The author may not actually hold the rights. 

If you have learned from the book and teach what you have learned then you are probably fine. But if you give them sections copied from the book to read, say over the internet, then you are probably not fine. 

Normally there is an expectation that the students have some access to the book even if they haven't purchased a copy. Borrowing a copy from the library or from a friend. So, an instruction to read Chapter 3 or do assignments 3.7 and 3.9 sends them to the copy for which they have access. None of that is copying and so is perfectly normal. 

But if you send them a copy of Chapter 3 or even the literal assignments then you are on shaky ground. Ethically and maybe legally. 

But you can make up assignments that are similar to those in the book, based on your own understanding. That should also be fine as long as you aren't being devious about just changing a value or two in a math exercise. 

I'm hoping that publishers will be a bit more lenient than usual, given the current situation. 

I'll note that copyright law varies widely around the world as does the definition of fair use. The creation of the Xerox Machine changed all of that and publishers have become quite stingy about such things. Your academic librarian almost certainly has guidelines about fair use in your country/region.