Currently there is no keyboard accessible alternative for reordering the task items which prevents assistive technology/ keyboard-only users from accessing the feature.

A number of implementations I've seen like this example from salesforce-ux enable user to press SPACE to move an item around with arrow keys and SPACE to place an item. Something to keep in mind for this element is making sure to not interrupt checking of checkbox (which can also be done SPACE).

Other references:

Upcoming WCAG change 2.5.7 Dragging Movements (AA)

4 major patterns for accessible drag and drop