It can be done in two steps.
First, add a new geometry column to you table:
SELECT AddGeometryColumn('my_table', 'my_new_geom_column', 4326, 'POINT');
Remember to change in the previous query my dummy values with the those that fit for database schema. The
4326 value is the SRID of the new geometry column, and if you don't make reprojection it must match the value of your existing geometry column.
In the second step, populate the new column with the desired values:
UPDATE 'my_table' SET
my_new_geom_column = ST_StartPoint(ST_ExteriorRing(ST_GeometryN(geom, 1)))
AFAIK Spatialite does not have a function to extract an specific vertex from a Polygon, but have it to extract vertex from LineString.
ST_ExteriorRing to get the "boundary" of the Polygon, a LineString and then
ST_StartPoint to get the first point of the exterior ring. But you can use also
ST_PointN(geom, vertex_number) (as @user30184 explains in the comments)
But as @raylight points in the comments if the geometry is a MultiPolygon with more than one geometry ST_ExteriorRing will return
NULL. It seams that ST_ExteriorRing in Spatialite only works for simple geometries:
-- With the example data provided by the OP
ORDER BY 3;
One way to avoid this is always get the "first geometry" of the collection/MultiPolygon
ORDER BY 3;
I'm not sure if there are more corner cases, where this approach does not really extract the first vertex of the Polygon.