The answer is no but actually sort of yes, because there are two relevant components to Social Security Act of 1935.

I suspect that when most people think of Social Security they are thinking of what was originally called Old Age Insurance or OAI (and later became Old Age, Survivors, and Disability Insurance or OASDI), created by Title II of the Act. This is a mandatory retirement insurance program for many workers that provides a monthly benefit upon reaching retirement age, but is only available to those who paid into the system (plus their survivors and dependents) and provides benefits based on the worker's contribution.

So OAI did not provide benefits to people who hadn't paid into the system. Such payments into Social Security began in 1937, but it was expected to take 5 years for there to be enough money in the fund to make the monthly payments. Contributing workers who reached retirement age before that time received one-time payments. (Ernest Ackerman, who worked for one day under Social Security, paid in 5 cents and got out 17 cents.) In 1939, Congress decided to start making the payments in 1940 (and added benefits for survivors and benefits).

This sounds like a clear no, doesn't it?

But Title I of the Act created the Old Age Assistance program or OAA. This provided federal support for state-run programs to support the needy aged people. These programs could not be conditioned on paying into the system, though states were free to have many other eligibility requirements such as being financially needy, being a resident of the state for a year (and 5 of the last 9 years), being aged 65 or older, and being a citizen of the United States. (They could not discriminate on the basis of race or locality.) OAA eventually spawned Medicare and Medicaid and was directly replaced by Supplementary Security Income or SSI (which united OAA with similar programs for blind and disabled adults).

So if we include OAA in the definition of Social Security--and we should, for it was perceived at the time as a major component of the Act alongside OAI--then the answer is that some needy elderly people did start receiving Social Security benefits even without paying in.