Can a triad marry?
In America, no, a triad cannot marry legally. In many states, it is illegal under bigamy laws to be legally married to one person and to “live as if married” with another person. Each state has its own laws regarding this, and if you are planning to hold a “commitment ceremony,” seek benefits like health insurance and child custody, or otherwise enter the legal maze of plural marriage, my recommendation is to find a lawyer in your area that specializes in this.
For more, check out “Legal Stuff” category of Polyamory On Purpose to see a breakdown of the bigamy/plural marriage laws in each state. The Legal page on the Poly Families website also has lots of resources for triads and other poly folks looking to marry.