From your post, it seems like in order to be eligible, you must be a direct descendant of someone who was harmed by city ordinances that discriminated against black people.
If that's the case, it would seem to be very difficult to argue that the city is engaging in racial discrimination: They're paying reparations for something that only affected black people so obviously the only people who can qualify are going to be black (to some degree), but they're only paying reparations to those who can show that they were harmed by their past policies. A recent Nigerian immigrant (or a black guy who moved there from the next town over) wouldn't be eligible, as far as I can tell.
It's not quite as cut and dried as that; there are a few huge legal hurdles this law must clear before it can go into effect, and it seems pretty unlikely it will clear them. Race is a suspect classification under the 14th Amendment, meaning that any government action that makes this classification must survive the strict scrutiny test. This is one of the most difficult tests for any law to pass, verging on impossible. The basic statement of the test is that the government can only use race as a classification if it is "necessary to promote a compelling government interest". That might not seem so bad, but the court has further broken down exactly what that means.
First, let's examine what a compelling interest is. The court hasn't exactly defined the term, but there's a strong implication that it refers to interests that are in a category of more crucial importance than those involved in ordinary government business. Saying it's important isn't enough; it has to be the kind of thing that is of such fundamental importance that it becomes necessary to suspend fundamental rights over. So the first step in analyzing the proposal is determining whether the interest involved is compelling, and to do that we must first identify the interest itself. So what is the government seeking to accomplish by writing $25,000 checks? At first glance, given the housing-centric nature of the proposal, that the government is seeking to provide redress for housing discrimination. One could argue that this is indeed a compelling interest, but there are two problems. The first is that is not meant to address any ongoing housing discrimination, or even any recent housing discrimination; the bill itself admits that the discrimination it intends to target ended in 1969, and while it does include provisions for those who have experienced discrimination since then, it requires them to provide proof of that discrimination. It's a stretch to say that the government has a compelling interest in addressing discrimination that happened over fifty years ago. The second problem the law runs into with respect to the compelling interest standard is that it does not identify any specific discriminatory policies or practices that it is trying to redress. The city claims that old zoning ordinances are the problem, but it doesn't get more specific than that. The Supreme Court ruled in Shaw v. Hunt that the government must identify the discrimination with specificity and establish a "strong basis to conclude that remedial action is necessary". Furthermore, the court has also ruled that "societal discrimination" isn't enough to overcome the standard. Therefore, these two deficiencies should prove fatal.
Second, we must unpack what "necessary" means in the test. This doesn't mean "necessary" in the sense that "If you want to go to that college it's necessary to get good grades". Sure, it probably helps to get good grades, and your only realistic shot of getting in might be because of grades, but there are probably other routes. You may get an athletic scholarship, or get into an arts program where grades are secondary to artistic talent, or have an uncle on the board of trustees, or be one of the few people who's lucky enough to get in with mediocre grades. "Necessary" means strictly necessary, as in "If you want to get into that college it is necessary to submit an application".
What this means as far as constitutionality is concerned is that the government must demonstrate that the only possible way to achieve the compelling interest is by making the suspect classification. Not that it's merely the easiest way, or least expensive way, or the most politically expedient way, but the only way. But before you can even establish that it is the only way to achieve the stated purpose you must first establish that it will actually achieve that purpose. Unfortunately, since we don't know what the proposal is specifically addressing, we can't determine whether it would achieve it's stated purpose. The proposal would allow families to use the money for such diverse purposes as assisting with down payments, assisting with preexisting mortgage payments, and making improvements to existing properties. Giving a young working-class couple assistance to make a down payment on their first home doesn't exactly achieve the same goal as giving an older middle class family a grant they will use to renovate their kitchen. As another user pointed out, the overall small number of grants that the bill provides for is unlikely to be sufficient to fully address whatever discrimination they are concerned about. Finally, there's no requirement that grant recipients even demonstrate that they have suffered personal consequences as the result of the historic discrimination.
The second component of the "necessary" requirement is that the framers of the law consider race-neutral alternatives. I won't dwell on this one as much since this is already a wall of text, but there's nothing in the bill to suggest the city looked at things like upzoning, or race-neutral housing subsidies and decided not that their proposal was simply the more effective policy, but the only effective policy.
Finally, I want to address a statement you made in your comment, namely:
They're paying reparations for something that only affected black people so obviously the only people who can qualify are going to be black (to some degree)
As I mentioned earlier, that's exactly the problem. If the policies in place prior to 1969 indeed only affected black people, there would be basis for suggesting that the response is appropriate. But they haven't even specifically identified what those policies were, let alone established that only black people were affected. If there were a specific ordinance that prohibited blacks from owning property in certain areas or interfering with their right to obtain mortgages or insurance, then okay. But I doubt this is the case. When pressed, I'd imagine the city would cite things like lot size requirements, or minimum square footage, or density restrictions, or other things that had the effect of making housing more expensive and thus out of reach of poor minorities, but similarly out of reach of poor whites. The law is pure posturing—it's a distribution of a trivially small aggregate sum packaged in a proposal that hits all the right rhetorical buttons. But it won't pass constitutional muster, and the chances that anyone who was actually harmed by past discrimination in Evanston will see any of this money is comically small.