Veteran Small Business Export Promotion Act This bill amends the Small Business Act to prohibit the Small Business Administration (SBA) from assessing a guarantee fee in connection with a loan made to a veteran or spouse of a veteran beginning the first fiscal year after enactment of this bill under the SBA's Export Working Capital, International Trade, or Export Express programs. If the President's budget for the upcoming fiscal year includes a cost for such a program that is above zero, this prohibition shall not apply to loans made during such upcoming fiscal year.