And if so, are there reputable places to obtain the coins (i.e. a local bank, US mint website, etc.)?
The US mint produces several gold coins. The two that are produced each year are the American Buffalo, which is 24k gold, and the (much older) American Gold Eagle, which is 22k. Both coins are bullion coins minted in specific fraction of Troy ounces (1 oz to 1/10 oz), although they also bear a face value, which is much lower than their metal value.
The mint produces these bullion coins in normal finish, and distribute them from a series of dealers. There are many large sellers of gold in the US, and they are generally competitive and reliable as long as you stick to those bullion (and not rare, antique, or so called investment grade) coins. Examples include Kitco, Blanchard, and Apmex.
The mint also strikes these coins in special proof, reverse proof, etc finish for collectors. It sells these coins both directly and through its network of retailers. Those coins are sold at a high premium over melt value, and are a very dicey proposition from an investment return standpoint.
The US mint also produces a variety of commemorative coins, some of which are minted in gold and silver. These gold coins are typically 22k, and contrary to the bullion coins above, they are minted in weights that mimic those of the old, pre-1929 circulation gold coins. They are not simple fraction of Troy ounces. Here is an example of a commemorative for 2011:
Yes. It produces American Gold Eagles.
It's unlikely you'll be able to get these at a bank. Coin dealers (online and offline), pawn shops, and eBay are good places to look.
I prefer the American Buffalo due to the higher gold content. You can purchase the proof version directly from the United States Mint.