It is generally though not absolutely illegal to use race as a basis for doing things. A private organization can, however, currently use whatever criteria it wishes, to decide who to give its scholarships to: UNCF has a list of about 25. Then the question is whether a demographic claim could be fraudulent (hence legally actionable). (There is a separate question regarding a school having a role in racially-restricted scholarships because there would be problems if the school gets any federal money or claims tax-except status, or engages in something that might be considered to affect interstate commerce, so let's assume that the flow of money is directly from the organization to the recipient).
There have been a handful of cases where a university faculty member has alleged a certain ethnic connection, which has been questioned, but as far as I know nobody has been fired for claiming an ethnicity that they aren't entitled to. There are two reasons. First, a university can't make race be a criterion for employment, so any putative fraud would not be material. Second, there is generally no objective proof of the falsity of such a claim (except: claiming to be a member of a particular tribe is a provable matter of fact – look on the tribal register).
In the past, when interracial marriage was illegal (and similar laws), it was necessary to legally define races, such as the Virginia Racial Integrity law, but such laws have been ruled unconstitutional and/or repealed. There are no legal standards of "racial purity" whereby a person can claim to be African-American only if [...]. On the contrary, there is increasing recognition that the individual has the sole right to self-identify, though especially for federal purposes one may be faced with limited choices such as those of the EEOC, which ask of the "category with which you primarily identify".
