Most applications for the competition are done online these days. When shipping your entries, here is how I do it:
First, find two boxes that nest inside each other. Purchase a roll of bubble wrap, a box of "snack" size baggies and a good tape gun with quality packing tape. Print your labels from the competition site when you register and cut them out. Keep your entries separate if shipping more than one entry. Be sure to read he competition guidelines for shipping and make sure you have a good address to ship to. To prepare your bottles for shipment, here is what I do...black out any identification markings on the cap, place your label inside the snack size baggie and seal it, fold over any excess and put a rubber band around the label. Now tear off a length of bubble wrap about 18 inches long. Roll each beer in the bubble wrap leaving an even amount of space on either end. Tape the wrap in the middle, fold over the top and bottom and tape securely. Do this for each beer. When you have them all wrapped, stack them together inside a good quality kitchen trash bag (in case one of them breaks or leaks) and seal the bag (tie it and tape it closed). Place the bag with the bottles in the small box, placing paper or more bubble wrap around all sides (I usually save packing materials I receive from online orders in a big bag for this purpose). Be sure to place your entry sheet in with the beer before you close and seal the first box. Put packing material on the bottom of the larger box. Set the small box inside the larger one and put packing material (paper, bubble wrap, etc.) around all sides. Then seal the box. Label it with the shipping address.
I get an account with UPS and fill out the shipping label online. This way I avoid having to explain what is inside to shipping clerks. I put the shipping label inside a large baggie and tape around the edges. Then all you have to do is drop it off at any place that accepts UPS shipments. All you have to do is tell them its all ready to go and just give it to them.
If you don't want to get a UPS account, then you simply take the box to a shipping center of your choice for UPS or Fedex and give it to them. They will weigh and measure it and ask you for an address. When they ask what's inside, I usually tell them it is "yeast samples double boxed". When asked about insurance, I tell them "less than $100" so you don't have to pay extra. And that's it.
It seems like a big pain, but I've never had any bottles break as far as I know. The snack size baggie prevents your label from running or smearing when they put it in the cooler to get cold and then take it out where it may get a lot of condensation on the bottle, depending on the temperature and humidity inside the competition area.
Cheers!