Have a look at FRIO. Their bags keep the contents at room temperature at really high external temperatures for ages (days!). It might be expensive mind you as you'll need at least two if not three bags for 14+spares syringes.
An alternative that I've used before is flasks. I put some ice cubes in the flask with about a week's worth of needles in a sealable freezer bag. (No. of weeks = no. of flasks required, depending on the size of the flask.) That keeps them at fridge temperature for up to 18 hours (I tested it!). Airport security do not like ice so you have to get it once you're in the departure lounge, but other than that, it's pretty straightforward. Not cheap either mind you, if you have to buy the flasks!
Remember that you don't need to bother keeping them cool apart from when the temperature is over 25 degrees so there's no need to chill them when you're at the airport or on the plane - Copaxone is fine at room temperature for up to 30 days.
Don't rely on using the plane's fridge / freezer - I've always been refused when I've asked :-(
Enjoy your holiday!