Changing a flat tire with missing safety lug nut key

Chances are the reason they won't sell you one is that they simply don't know which one you need, hence their asking you to take it to them, so they can compare to find the right key.

Another option might be to get a single use universal key - I suspect these aren't available to consumers, but a local tyre dealer might have them. Effectively they're a key made of a much softer metal that is hammered onto the nut and forms the right shape as it does so, allowing you to remove the nut. It does damage the nut though, so you'll then need to get a new set - which will come with a reference number in case you ever need another new key!

This happens occasionally. You can purchase a kit to remove the locking lug nut.

https://www.amazon.com/Wheel-Lock-Removal-Kit/s?k=Wheel+Lock+Removal+Kit

There are lots of different types here. Typically you use a sledge hammer to place the removal tool over the top of the locking lug nut. Then use an air hammer lug wrench to remove it. The obvious alternative is to hire a tow truck to bring the car to a repair shop that already has those tools.

Note, depending on the wheel shape, and the type of locking lug nut, it might be possible to use a simple ball peen hammer and cold chisel to remove the lug. You need plenty of clearance room. I've removed truck tires on deep mag wheels that way, but I was pretty careful. Took a long while, but it worked. But newer wheels don't really have that easy access shape. I'm assuming your rims are cast aluminum and not stamped steel.

Good luck with the repair.

The special lug nut is an anti-theft device, to prevent your tire from being stolen by anyone with basic hand tools. There are many different shapes, you have to know the code to buy a replacement, otherwise anyone could buy one and steal your tire.

As for getting going try re-inflating the tire, if it's a slow leak you may be able to do a short drive. If it won't inflate you can get a bottle of quick fix style instant tire repair which comes in an aerosol can, this stuff has gunk which can plug some holes, and gas to pump up your tire. Keep in mind the tire can't be repaired afterwards, and the gas in the tire is flammable so needs to be deflated with care.

You could also call a mobile tire service if they exist in your area to see if they could get it off for you.