If you've started a nonprofit and only you and your wife are the only ones running it with no employees ....?

Yes.  The IRS recognizes two kinds of TINs (taxpayer ID numbers):   Social security numbers and EINs.  Those are the only two choices.   SSN doesn't work for your entity because it's not an individual person.

Right from the IRS page for EIN numbers:

Purpose of an Employer Identification NumberEmployer Identification Numbers are issued for the purpose of tax administration and are not intended for participation in any other activities (e.g., tax lien auction or sales, lotteries, etc.)

Exempt Organization InformationIf you believe your organization qualifies for tax exempt status (whether or not you have a requirement to apply for a formal ruling), be sure your organization is formed legally before you apply for an EIN. Nearly all organizations exempt under IRC 501(a) are subject to automatic revocation of their tax-exempt status if they fail to file a required annual information return or notice for three consecutive years. When you apply for an EIN, we presume you’re legally formed and the clock starts running on this three-year period.Example: Your organization applies for an EIN in November 2014 and chooses a December accounting period. Your first tax period would end on December 31, 2014, and your first return or notice (if your organization does not meet one of the few exceptions to the annual reporting requirement) would be due May 15, 2015. You would be subject to automatic revocation of your exemption if you fail to file for the three periods that end December 31, 2016 (return/notice due May 15, 2017) or for any consecutive three-year period thereafter.Related TopicsState and Federal Online Business RegistrationOnline EIN: Frequently Asked QuestionsEmployer ID NumbersSystem RequirementsPrivacy Act Statement and Paperwork Reduction Act NoticeBusinesses with Employees

You need one.

The EIN is used for other stuff too.

A non-profit is required to be incorporated and all corporations must have an EIN. The term is used interchangeably with TIN (taxpayer identification number). As a corporation, you and your wife could be employees of the corporation. There is a whole section on www.irs.gov dedicated to charities and non-profits. Read the requirements for becoming a non-profit.

You literally don't know the first thing about operating a nonprofit corporation. Why do people do this? It's like jumping into a plane, somehow getting it to take off, then saying, "How do I fly this thing?"

What could go wrong?

This will be nothing but a huge headache for you until you hire someone who knows what they're doing to operate and comply with tax laws.

If you are receiving funds ( I assume you would be because you used the term non-profit) you need to obtain tax-exempt status from the IRS. Complete IRS Form 1023, Application for Recognition of Exemption Under Section 501(c)(3) of the Internal Revenue Code. Completing this form can be a daunting task because of the legal and tax technicalities you'll need to understand.

A non-profit is a separate entity, and needs an identification number for government purposes that is not the personal Social Security number of either of you. What it is called is irrelevant. They call it an EIN number rather than a separate category.