Accelerate Our Startups Act of 2017 This bill amends the Small Business Act to direct the Small Business Administration (SBA) to develop and implement a grant program for state and local governmental and other nonprofit organizations located in the United States that support startup businesses by providing assistance for construction costs, space acquisition, and programmatic purposes. A grant recipient must: (1) demonstrate that it will use the grant to provide assistance to at least five client businesses per year that have been in business for less than five years, (2) require client businesses to pay an entry fee to participate in the program, and (3) submit annual progress reports. The SBA shall: (1) establish grant criteria favoring recipients that provide startups with office, manufacturing, or warehouse space, access to capital and professional services, and a formal structured mentorship or developmental program that assists with building business skills and competencies; and (2) in determining whether to make a grant, take into account promoting growth in underserviced areas with sufficient population density and the organization's entrepreneurial leadership experience, ability to utilize and leverage local strengths, and record with regard to client business participation. The SBA must condition receipt of a grant under this bill on the recipient obtaining matching funding for the same purposes from a local or state government grant and nonpublic funding.