You do not need to do anything. You will receive your stimulus the same way you receive your SSI.
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I'm sorry, but unless the IRS has made an agreement with the Social Security Administration in the past day, that is misinformation. SSI is NOT the same as SS and SSDI (disability), and SSI does not receive a 1099 tax form each year. While SSI is not required to file taxes, the IRS has said that people on SSI and those who receive VA benefits must register for the stimulus check using either their nonfiler tool or the one here on Intuit/Turbotax.