I doubt you are receiving SSI, Supplemental Security Income, which is a welfare program for disabled adults and disabled children.
You are probably receiving Social Security Survivor benefits from your deceased husband. Social Security is not a welfare program and for survivor benefits, you have to meet certain age and relationship rules, but no disability rules, unless you are a widow over age 50.
If you and your children are receving Social Security Survivor benefits and not SSI like you state, you will be occasionally asked to complete a simple form about how you spent the money. You should have an idea of how much of the money for the children goes to household expenses and how much goes to personal expenses. You generally don't have to supply detailed records, as long as your answers make sense.
You should probably go to your local office or the ssa web site (socialsecurity.gov) and obtain brochures or read material on your duties as representative payee. Then ask questions.
Your personal survivor benefits last until the youngest child is age 16 or when you start earning over the exempt amount from your job. Your children's benefits last until they turn age 18 unless they are still in high school on their 18th birthday. Then, benefits last until age 19 or high school graduation, whichever comes first.
You can reapply for reduced benefits as a widow at age 60 if you remain unmarried. You can reapply for reduced benefits at age 50 if you remain unmarried and are disabled. Or you can reapply at full retirement age (somewhere before age 68) for unreduced benefits.

