Yes, you can definitely "tame" a wild mustang.
The ones you see wandering around almost certainly belong to the BLM (Bureau of Land Management). You could get into a lot of trouble for "taking" them. At the same time, it is very easy to adopt one through the BLM. They have regular auctions.
If you are not up to the task of training one yourself, a good idea is to go to a "Mustang Challenge" or "Mustang Makeover." Both are run by the Mustang Heritage Foundation, in association with the BLM.
The challenge basically takes 50-100 trainers, and gives them each a Mustang. They have 100 days to train the horse. After that, they come to a central location and compete to see who did the best. At the end, all horses in the competition are auctioned. Some of the trainers do bid on their own horses, but many do not.
A trainer I work with participated in mustang challenge this year. Not only did he get his horse at auction, but I and another friend each bid on and adopted mustangs from the challenge. They are basically green broke 3 year olds, but that "green broke" training makes all the difference in the world if you are not a professional trainer.
All the mustangs I have seen (from that competition) are VERY calm, level headed, laid back horses. They are not "wild" or crazy. They are very smart and quick to learn. Most seemed to bond very well with people..
For what these horse were doing in the competition, the price is amazingly low. This is the best value in the horse world - as long as you understand you are dealing with a young, green broke horse. These are not 12 year old, been there done that, bombproof horses. The advantage of that is you have a solid foundation and can train them the way you want from there.
Check out MUSTANG HERITAGE FOUNDATION : OUR LAND, OUR HORSE
for more info.