While some states allow these kinds of marriages, others do not. In general, laws about whether it's legal to marry your cousin fall into three categories:
States that allow cousin marriages. This is the situation in 19 states, including big ones like California, Florida, and New York, along with the District of Columbia, according to the National Conference of State Legislatures.
States that prohibit first-cousin marriages. Tying the knot with a first cousin is not allowed in 25 states, according to the NCSL.
States that allow first-cousin marriages under certain conditions. Five states allow first cousins to marry if the cousins are of a certain age (50, 55, or 65); four of those states also allow cousin marriages if one cousin is unable to reproduce. In Maine, first-cousin marriage is allowed if the couple receives genetic counseling by a physician.
But there are even more nuances to state laws. Some bar first-cousin marriages, but may allow marriages between second cousins (i.e., the children of first cousins), half-cousins, and adopted cousins.
State criminal laws against incest may also effectively prohibit marriages between cousins. That's why it may be a good idea to consult an experienced local family lawyer for advice on whether it's legal to marry your cousin where you live.
Note that the US also still has a patchwork of restrictions on same-gender marriages. No state currently recognizes polygamous marriages.
What we need is nationwide full marriage equality. A woman should be free to marry another woman, even of that woman is her cousin, and even if she is already married, as long as all involved consent.