I don’t know any reason why a donor would be legally required to provide a physical address if the donee does not require it.  Failure to get a physical address may create some problems for you, however.

The IRS says in the instructions to Schedule B, which requires a list of significant contributions from many charities, that the Schedule B may identify a donor as anonymous and omit the name (and presumably the address) only if the organization does not know the donor’s identity.  The IRS obviously recognizes that anonymous donors can exist and does not require the charity to disclose a name or address it does not have.  But you would be required to disclose the name and email address if that is all that you know.

You are more likely to have an issue with state charitable solicitation registration.  39 states and the District of Columbia require charities, other than those excluded or exempted, to register before they ask a resident of the state for a contribution.  If you are not registered in all the states, you could get into trouble if you get donations from any of the states in which you aren’t registered.  Even if you didn’t solicit the first contribution, you won’t be doing your job if you don’t ask for more.  Therefore, you really need to know where your donors are located to know where you should be registered. 

Ask for the physical address on your forms.  It will eliminate a lot of uncertainty.