What's the purpose of syrup in toffee apples?
The bane of sugar syrup or caramel making is unwanted crystalization. A few stray sugar crystals, a premature stir, and your caramel gets grainy instead of smooth.

Corn syrup is an invert sugar (glucose), which can prevent this. Alternatively, a bit of acid (a few drops of lemon juice, a pinch cream of tartar...) will break some of the sucrose (plain sugar) into fructose and glucose. 

Do you have to use it? No. It's mostly a safety net. If you follow proper procedure, like not stirring, washing down stray sugar crystals from the walls of your pot..., you can make your caramel apples without syrup. And there are recipes without syrup to be found on the Internet, for example here at Serious Eats.