It might be possible.
(1) You'll want to set the elements representing the vignette, likert questions etc. to /erase=false so the remain on-screen across trials.
(2) You'll need at least two different <trial> elements. One to collect the likert responses (you cannot use the <likert> element here; you'll have to use standard <text, <picture> etc. elements to build them), one to display and collect response to the dot.
(3) You'll have to set up various <values> and <expressions> as well as make use of various timing properties (elapsedtime, latency) to calculate response times.
(4) During "dot"-trials, you'll may not be able to respond to the likert question (and possibly vice versa).
(5) You'll have to work out a relatively complex timing scheme for the dot flashes and use /timeout and/or /trialduration to implement the scheduling (alternating between vignette response and dot response trials).
All in all this requires a somewhat advanced understanding of Inquisit's programming model. I would thus recommend you first get down the basics by completing the various tutorials in the documentation (if you haven't done so already). Then move on to study a selection of scripts from the Task Library which illustrate more advanced programming approaches and techniques.
The overall limitations you'll have to cope with are
(1) You cannot run multiple <trial> elements concurrently (as would be required for a true "concurrent" dual-task), only one at a given time.
(2) A given <trial> element only allows input from a single input device, i.e., you cannot have it collect mouse (clicking a likert scale) AND keyboard responses (space bar press).