The rules for immigrants are the same as for everyone else. Immigrants who filed their taxes in 2018 with social security numbers, or who filed or can file taxes in 2019 may be eligible. If you or your spouse file joint returns and either of you has an ITIN, you will not get the payment. A dependent child with an ITIN does not qualify for the $500 payment.
Immigrant workers who file Form 1040NR will not get the payment.
Does Immigration count the stimulus payment as a benefit for the public charge test?
No. The stimulus payment is a tax credit and not a public benefit.
What if I just received my social security number?
If you or your dependent child age 16 or under just received a social security number and you already filed your 2019 tax return without it, you can file an amended return to get the stimulus payment.