Old Town is not actually in Kissimmee, it's in the Maingate area which is closer to WDW than Kissimmee, and WAY closer than the Amtrack station. Cab fare each way should run less than $15 from Boardwalk.
However, if you want supplies, Old Town is not the place to go - there are no grocers or drug stores there, it's all tourist souveniers and a few fast food eateries.
For supplies such as snacks and drinks, try
WalMart or Walgreens. You can get milk, soda, chips, pretzels, Pop-Tarts, donuts, and other snacks at either. For variety, go to WalMart, for speed (because the store is smaller) go to Walgreens. There is a Walgreens on SR 535 one intersection north of the Crossroads shopping center, and a WalMart on SR 535 about 2 miles south of the Crossroads.
Many people will tell you to go to the Goodings supermarket at the Crossroads. Gooodings is a nice store and has everything you need, but the prices are outrageous. If you don't mind paying twice as much for the convenience, Goodings is a good choice.