Most high altitude balloonists are using latex weather balloons to fly their projects. These can be either purchased new, or sometimes are available as surplus equipment. The two main brands are currently available are Totex/Kaymont and Hwoyee.
Kaymont is the supplier in New York that has been supplying weather balloons to the government as well as hobbiest industry for years. The Kaymont balloons are well known among the balloonists, and their behavior is predictable and well-documented.
Hwoyee balloons are relatively new to the hobby community, and can be purchased online from suppliers such as Scientific Sales, Inc. They are generally a good balloon, although some groups have reported unpredictable burst altitudes (both higher and lower than expected).
Balloons can also be purchased in the surplus market. Watching eBay will usually yield some balloon results from time to time, although some care should be given in that latex balloons can be mis-handled, and they do have a finite shelf life.
As a side note, balloons are sold by the mass of the latex used in their manufacture. Historically, the 1200g (gram) balloons were the most popular and they will regularly hit a burst altitude around 100,000'. If you can live with a lower burst altitude (around 70,000'-80,000'), a 600g balloon may suffice, especially if you have a very light payload (less than 3 pounds). The larger balloons (2000-3000g and up) are typically reserved for very large payloads (over 10-20 pounds), or for very special case applications.