Governor Chafee’s spokeswoman, Christine Hunsinger, confirmed this morning that one of the items in the FY2013 budget was a tax amnesty program. It will run from September 2 to November 15, 2012 and will apply to state taxes including income, sales, use, and unemployment insurance. Note that while monetary penalties and prosecution will be waived (for qualified applicants), interest would still be owed, albeit at a reduced rate.
More details from the CCH division of Wolters Kluwer Law & Business.

The program provides amnesty from penalties and from civil or criminal prosecution for any tax imposed under Rhode Island law and collected by the tax administrator for any taxable period ending on or before Dec. 31, 2011.
Amnesty will be granted only to those taxpayers who apply on or before Nov. 15, who have paid the tax and interest due or who have entered into an installment payment agreement as a result of financial hardship.
Amnesty will not be granted to taxpayers who are under any criminal investigation or are a party to any civil or criminal proceeding pending in federal or Rhode Island court for fraud in relation to any state tax imposed.

As for how to apply, there appears to be nothing yet on the RI Division of Taxation’s website; it is early. I left a message with an official at the Division of Taxation asking him how interested taxpayers would get started. When he calls back, I will post an update.
UPDATE
The Division of Taxation has advised that, as the start date of the program approaches, information will be made available on the Division’s website for interested applicants.