It's possible to add on-page anchors to blocks with WYSIWYG fields.
The first step is to add a link with your anchor name as the URL. For instance, the example below where the anchor name is #anchor:
<a href="#anchor">Example Link</a>
Then, the content you want to link must have an ID with a value that matches the anchor name, which in this case is #anchor:
<h2 id="anchor">Example On-Page Destination</h2>
Note: This solution may conflict with pages that have tabbed layouts because the tabs are also dependent on anchor links. However, this solution should work for pages that do not have a tabbed layout.