How to export shapefile to SpatiaLite

Several options!

1. Within QGIS, you can "File|Save as" the shapefile directly as a spatialite database; the limitation here is that you can make a database with only one shapefile.

2. You can use the command line spatialite_tool to load and manage shapefiles (tutorial here).

3. It is really worth exploring the spatialite-gui and spatialite-gis standalone applications, which let you view and modify data directly. In the gui app, create a new database (second toolbar button), then load a shapefile (green world icon with arrow) - you must know and input the SRID of the shapefile in the resulting dialog. The advantage here is that you can easily add multiple shapefiles, and build a large database.

Yes it can be done with QGis.
Using the QSpatiaLite Plugin

"Manage your SpatiaLite databases within QGis"

After installing the plugin you can run it.

You will receive a series of messages/errors stating that you do not have a spatialite db.
it will step you through getting one installed (choose a location for the file) and finally converting it to spatial.
Then in this dialog you can import your QGis layer.

Then: you don't really need this. It only works to create a new db.
From the QGIS menu
Layer | Save As...

From the QGIS 'Save vector layer as...' dialog box
Choose Spatialite from the Format drop down list.

You can right-click the layer and select "save layer as..." and choose spatiaLite as your format. Your only option there is to create a new database, i.e. you cannot add it to an existing database.

The QspatiaLite plugin adds a lot of functionality to QGIS, so give that a try as well.