You most certainly can purchase a house at any age. At age 16 it would be best to purchase it within a trust fund which allows a parent or guardian to manage the fund until you turn 18, but you would still be the legal owner.
However, in the USA minors under 18 cannot be held legally responsible for a contract such as a mortgage loan. Therefore no bank would ever loan you money toward the purchase until you are 18, and then you'll need a few years of good credit and steady income.
So if you have the cash to buy a house, then go right ahead.