Are there any legal issues in having someone's book as the basis and as a textbook for an online course?
I cannot see any problem with this as long as no copyrighted material is accessible in such a way that it breaks the copyrights  of the book. Although the outline of a book probably can be considered the intellectual property of the author (and possibly the publisher depending on copyright), I doubt anyone would object since the purpose of writing a (course) book is to provide a product to support teaching of a specific topic. The structure of a book can often be the most logical way of presenting the material. The key issues is rather if you reproduce parts of the book such as providing the exercises on-line. As long as participants must obtain the material in the book on their own you should not worry. From this perspective it is no different from a normal university course (where it is also illegal to copy and distribute material freely from a book (or equivalent).

So for an on-line course, you just need to be very careful with reproducing text or figures from copyrighted materials in ways that breaks the rules. You can also contact the holder of the copyright (author/publisher) to obtain the right to provide parts within the course. The success of this may depend on the system you use for on-line material and what file formats you use for distributing the materials. Such details will likely be made clear by the copyright owner if you ask.