Vanity has slots for up to 4 slides and their links in the banner. Here is how to increase it (I will take it up to 6 for this example).
1. Open Theme > Template Editor and edit settings (in the Preset Settings section at the bottom)
2. Scroll down until you see where the slideshow images are set. You will need to copy and paste new rows, up to the desired number of slides, and increment the number in the name attributes. For your convenience, Vanity ships with an 4 additional slides already copied, but commented out, so you can just move the comment line down between rows to get more.
Save and close your settings.
3. The final piece of the puzzle is to update the index template. At the top you will see a variable - change that to your new number of slides.
Save and close index.liquid.
That's it! If you visit theme.settings you will now see additional slides (and their links).