Veterans' Right to an Attorney Act of 2003 - Repeals Federal provisions: (1) prohibiting a fee from being charged, allowed, or paid for attorneys' services with respect to veterans' benefits claims at any time before the date on which the Board of Veterans' Appeals first makes a final decision in the case; and (2) allowing such a fee agreement only in cases arising with respect to veterans' housing and small business loans. (Such repeals in effect allow veterans to pay fees for attorney services during any stage of the claims process.)