With SQL Server, you can open Management Studio, type in some SQL code containing a BACKUP DATABASE statement and execute it to backup the database. The SQL Server documentation can show you the appropriate SQL code. In VB, you can create a SqlConnection and a SqlCommand containing that same SQL code, then Open the connection and ExecuteNonQuery the command.
I don't use MySQL but I would assume that the situation would be similar. Go to the MySQL documentation and find out what SQL code you can use to backup a database using a management tool. Once you've tested the code and know it works, create a MySqlConnection and MySqlCommand to execute the same SQL from your VB app. You'll need to download and install Connector/Net from the MySQL web site if you haven't already.