Nonprofits, like for-profit businesses, are usually considered places of public accommodation under our civil rights laws, and are allowed to pick and choose their customers on any grounds they want so long as it does not constitute discrimination against a legally protected class, such as race, creed, sex, or national origin.  Giving preference to veterans or first responders would not be a violation of civil rights laws, especially in view of the increase in the number of women who are veterans and first responders.

Not every organization has the capacity to handle all of the requests for its services (think selective colleges and universities, for example) and they have to set the ground rules for selection.  It could be first-come first-served, but giving preference to veterans and first responders, who some would think are particularly deserving because of their record of service to others, is not an irrational or illegal standard for them or their donors.

I suspect that if you read the organization’s website closely you will find some reference to their preferential policy.  And I think it is admirable that they didn’t mislead you when you sought to apply.  Unfortunately for those who can’t be served, if they don’t have the resources to help everyone who applies, there is no legal requirement that they do so.