Although adopted siblings wishing to marry is an extremely rare and unusual situation, it is a possibility. But is it legal? The short answer is "no." It appears that throughout the U.S. and in most of the world, parties that are direct descendants or siblings—including adopted (by law) relationships— are not allowed to marry. In some rare instances, however, there may be exceptions.
In most locales, it is commonly believed that when two individuals share the same parents—regardless if one has been adopted—they are full siblings.
One state that leaves this issue a bit open-ended is Colorado. The wording of the Colorado statute regarding prohibitive relationships does not include the word "adopted," so adopted siblings who want to marry one another should follow up with a county clerk in that state.
The Definition of Incest
Marrying a sibling is considered incest—Dictionary.com defines incest as "sexual relations between closely related persons." There is no specification about blood relation in the definition, leaving the question of adopted siblings unanswered. Some countries view adopted siblings as full siblings nonetheless and see any sort of relations between them as incestuous.
The article Laws Regarding Incest in the United States states that the laws vary widely between jurisdictions in regards to both the definition of the offense and the penalties for violating the laws. Therefore it is necessary to research the particular state's rules and regulations and which states prohibit marriage between adoptive and blood siblings.
The Religious Perspective
Different religions have different perspectives on an adopted sibling and blood sibling marrying each other. The Church, according to canon law, views an adopted child the same as a child from the family's bloodline. Canon 1094 states that those who are related—even by adoption—cannot obtain a legal marriage contract.
This prohibition exists to support any civil laws as well as psychological stability. Because this is an ecclesiastical law, it is allocated by the siblings' local bishop who could make an exception if, for example, the adopted child was brought into the family after the sibling moved out of the house.
In Judaism, however, there isn't that specific restriction. Although it is discouraged, since the adopted child isn't part of the family's hereditary blood, he or she may marry a sibling from their adopted family. It may not be considered incest, but it is considered unseemly.
According to Islamic law, adoption is prohibited and therefore Islam does not have any laws on the matter.
The Reason for Prohibition
The assumed reason for these laws prohibiting marriage between adopted siblings is to prevent creating perceived "family chaos." It is likely that your parents would loudly object to such a union. It is a much better idea to never enter into a romantic relationship of any sort with an adopted sibling, even if this person came into your life later in childhood.
The bottom line is that no siblings, whether by blood or adoption, can legally marry—nor should they.