Arrange a rack in the middle of the oven and heat to 350°F. Line a rimmed baking sheet with aluminum foil. Arrange the saltine crackers in a single layer on the baking sheet.
Place the butter, sugar, salt, and vanilla in a 2-quart saucepan and bring to a boil over medium-high heat. Reduce heat to medium and cook, stirring just a few times, for 3 minutes. The mixture will be thickened slightly and homogeneous.
Working quickly, pour the hot toffee mixture over the crackers, spreading it so most of the crackers are completely covered.
Bake until the toffee is bubbling and both the crackers and toffee are beginning to brown, about 15 minutes.
Remove the pan from the oven and immediately sprinkle with the chocolate chips. Let the chocolate melt for 5 minutes, then use a spatula to spread the chocolate into an even layer. Set aside to cool for at least 2 hours before cutting into 30 pieces.