You are correct that payments received from a charitable organization by victims of a disaster are considered gifts and not taxable income.  But you don’t say who “we” is, how much you think you might raise, and why you want to make the gifts directly. 

If you are a group of individuals or an entity that is not already a public charity, you will probably be more effective if you work with an existing charity to raise funds and help administer the program.  If you have to start your own organization, you will have to file with the IRS for recognition of exemption, which will take many months before you obtain your exemption letter.  In the meantime, you will have a much harder time raising contributions for the program.  Unfortunately, there have been a lot of phony groups that have scammed the donating public in prior disasters and a lot of people are wary of pop-up organizations soliciting for disaster relief purposes.

If you are already a public charity, the keys to operating the fund are that it be open to a broad range of individuals in need of assistance because of the disaster, that you base your assistance on the particular long term or short term needs of the recipients, and that you document the basis, the amount and type of assistance that you provide.  You can find out more about the specific requirements in IRS Publication 3833 on disaster relief.  If you are a public charity that has not done this type of disaster relief before, you will have to notify the IRS that you have undertaken a new activity when you file your Form 990 return for this year.  You may also need to change your state charitable solicitation registrations if such activity is not contemplated by your existing registration.

It may look like a simple and important activity to undertake, but if you haven’t done it before, it is not a simple task to do as a start-up.  You will probably be a lot more effective if you work instead with people who have experience.