In a small saucepan, stir together the sugar, lime juice and water. Bring to a boil, then reduce heat and simmer for about 8 minutes or until mixture is reduced by just about half. You'll have to eyeball at first. You'll know you are close when the bubbles start looking white and fizzy. Also, you don't have to be super precise about how much you reduce it, but to be sure you've used the right amount you can scrape it into a bowl and weigh it. The starting weight is 187 grams and target weight is between 85 and 115.
So you can weigh it, or skip that, but do scrape it into a bowl and allow it to cool. It will also thicken quite a bit. If you've reduced it quite a bit (85 grams) it will be more golden in color and so thick you'll have to use a blender for the next step. If you reduce it more gently and stop when it's down to about 112 grams, it will have a slightly cleaner lime flavor and will be easy to just whisk with the coconut milk.
Pour the coconut milk into a blender container and scrape in the cooled thick lime syrup. If your syrup is thin enough, just whisk the two things together in a bowl. Add lime zest and vanilla and blend or stir until creamy and homogeneous. Scrape into a bowl, then cover with plastic wrap and chill for 2 hours or until very cold.
When ready to process in the ice cream machine, whisk in the vodka. Scrape into the ice cream maker and process as per the manufacturer directions.
Scrape ice cream into a pint container layering with crushed graham crackers or gingersnaps.
Freeze for at least an hour or until you are ready to serve. Ice cream will still be fairly soft at this point. Otherwise, freeze overnight and serve when convenient. Ice cream will freeze solid, so allow it to stand at room temperature for about 10 to 15 minutes or until it softens. It will go from icy to creamy.