If you are covered under the health care law then you do not have to pay the penalty of not having a social security number. Unfortunately, you are not able to enroll in a market plan to replace or supplement your medicare coverage.
You may be able to get Medicare while you wait. There are 2 ways that you can apply for Medicare, one is creating an account or log in to complete an application and you should say "yes" when you are asked if you have disablity. This will automatically forward your application to your state medicare agency when completed.
If youre eligible for Medicaid your Medicaid eligibility can continue even after you enroll in Medicare, but if you are turned down for Medicaid you can enroll in a private health plan through the marketplace.
If you do not have a disability or do not qulaify for disability benefits and need to get health coverage, you have a few options inside the Health Insurance Marketplace.
You can create an account or log in and fill out the marketplace health coverage application to find out if you could qulify for a private health plan with premium tax credits and other savings based on how much you make and the size of your household.
The fee for not having coverage: under the health care law most people must have health coverage or pay a fee or get any exemption from a fee, which applies no matter of disability status. If you do not have coverage through Medicare or Medicaid, another public program, a job, the Marketplace, or another source, you may have to pay the fee. You can get an exemption from the fee if you have low income, hardship or others factors.
Depending on your situation you may be eligible for savings based on how much you make. But there is one catch, if someone claims you as a tax dependent you can buy a plan through the marketplace but can not qualify for savings based on your income. But you can pick a "Catastrophic" health plan, which is a way to protect yourself mainly from worst case scenarios. If you are in school you may be able to enroll into a student health plan and meet the standards for having coverage under the health care law.
Under the health care law you must qualify health coverage or pay a penalty, or pay a penalty on your next federal tax return. In 2017 the penalty was 2.5 % of your total household adjusted income or in other words $695 per adult and $346.50 per child, but in 2018 the penality has not yet been announced but is supposed to increase.