Neons can sometimes become nippers, especially when kept in less then the minimum for a proper school(5-6), but often still in a smaller school they'll get bold and start nipping after a while. But a lot of people also keep them with bettas very successfully, so its really a hit or miss.
Backup plans are always a good idea. ;)
Anyway, if you really can't order online/have a parent order for you, do some google searching for an Ace Hardware around your area.
If there is absolutely no way for you to get your hands on some pure ammonia...you could either take the risk of the shrimp/fish food method and jut hope it works out, or you may have to do a fish-in with the fish you already have. Provided you keep up with the water changes and test once or twice a day, your betta should be fine, but neons are sensitive....they don't do well in unestablished tanks.
If you end up having to go with fish-in, add in some hardy live plants if you can to help a little with the water quality.
Regardless of the cycle method you end up with, it may not hurt to give a bottled bacteria like Tetra SafeStart or Seachem Stability a try. Its a bit of a hit or miss, but I've head of several accounts where they have helped to boost the cycle a little for people.