stored procedure case statement
I think this is a better, safer, more self-documenting, and more scalable approach:
IF @ApplicationType IN ('Clinical','Patients')
BEGIN
  SET @whereClause += N' AND vpgl.AppType = @ApplicationType';
END

IF @ApplicationType = 'Web'
BEGIN
  SET @whereClause += N' AND vpgl.AppType IS NULL';
END

...
SET @sql = @sql + @whereClause;

EXEC sys.sp_executesql @sql,
     N'@ApplicationType varchar(32)',
     @ApplicationType;

If you really, really want to use a CASE expression (which is not meant for control of flow), you can say:
SET @whereClause += CASE 

   WHEN @ApplicationType IN ('Clinical','Patients')
   THEN N' AND vpgl.AppType = @ApplicationType'

   WHEN @ApplicationType = 'Web'
   THEN N' AND vpgl.AppType IS NULL';

   ELSE N'' END;

...
SET @sql = @sql + @whereClause;

EXEC sys.sp_executesql @sql,
     N'@ApplicationType varchar(32)',
     @ApplicationType;