According to the instructions for Rhode Island Form RI-1040H Rhode Island Property Tax Relief Claim, the property tax relief claim must be filed by April 15, 2013. However, an extension may be allowed at the Rhode Island Division of Taxation’s discretion for sickness, absence, or disability.

According to the Division of Taxation, there is not an actual "extension form" to file. You should send in a letter with Form 1040-H explaining why you are filing the form late. Based on that letter, the Division of Taxation will decide whether or not you will receive the credit. 

Rhode Island has the taxpayer send a letter explaining the situation instead of having an actual form to file for an extension because the "extensions" for the property tax relief are handled on a case by case basis. Please note that filing Form 4868 before the due date of the return is NOT sufficient to extend Form 1040H.