Yes. Tax-exempt employers such as 501(c)(3) organizations are eligible to offer 401(k) arrangements.1 State and local government employers (including political subdivisions and agencies thereof) generally are prohibited from offering 401(k) arrangements to their employees, but certain rural cooperatives may do so.2
1. IRC § 401(k)(4)(B)(i). Of course, 403(b) plans are an option for nongovernmental tax-exempt employers and are also available to governmental tax-exempt employers, like school districts.
2. IRC §§ 401(k)(1), 401(k)(2), 401(k)(4)(B)(ii).