Under existing regulations, higher-rate taxpayers are able to claim for themselves the difference between the basic and higher rates of tax on donations if they fill in a form. Many then choose to pass this on to charities.
But the institute says half of those donors who are eligible to claim the rebate are not filling out the forms because they are too complicated.
The petition, which was published on the 10 Downing Street website this week, calls for changes to allow donors to assign the money directly to charities, although they would be allowed to keep the relief for themselves if they wished.
Simplified rules would help smaller charities, according to the institute, because the current system means few people claim back higher-rate tax relief on anything but the largest donations.
Louise Richards, director of policy and campaigns at the institute, said the petition would demonstrate to the Government the "groundswell of support" behind the campaign.
"It is my strong belief that these changes will be received very warmly by organisations," she said. "That is why it's important to show the Government the high level of support on this."
The institute is calling on its members to sign the petition before the end of November to increase pressure on Chancellor Alistair Darling to include the changes in his pre-Budget report.
Richards said the institute was also continuing to lobby for an 'opt-out' Gift Aid system in place of the current 'opt-in' approach.