You always owe a correct tax return, even if you don't have a social security number and aren't legally allowed to work. (The IRS just wants their tax money and won't snitch on you to ICE.)
Normally, you would file your tax return by including a form W-7, which is an application for an ITIN, international tax ID number, which is used by non-resident aliens and resident aliens who don't have SSNs.
However, if you are eligible for an SSN and have applied, then my advice depends on whether you owe tax or expect a refund:
- If you expect a refund, there is no penalty for filing late, and your refund will be paid if you file up to three years late. If you are eligible for an SSN and are expecting your SSN soon, I would wait until you get it, and then use it to file any prior year tax returns you are required to file.
- If you expect to owe tax, then there are penalties for filing late, which accumulate over time. To minimize those penalties, I suggest filing now, by including a form W-7 ITIN application with your tax return. You will have to print your return, and you mail the return and the W-7 to the address for ITIN processing. After the IRS issues the ITIN, they will process your tax return. Then when you get the SSN, you will start using it in the future, and you will send a letter to the IRS informing them of your old ITIN and new SSN so they can merge your records.