Social Security Questions? Ask a Social Security Expert
Yes, you can. If your child is/was born in a US hospital they will provide you the paperwork to complete and they will submit it for you. That is a requirement of law.
However, if your child is not born in a hospital (home birth), you will need to contact Social Security for the paperwork. Use www.ssa.gov for this.
If you are not a US citizen you cannot be issued a Social Security number. Instead you can be issued a Taxpayer Identification Number (TIN) which serves the same purpose. Here is a link to the IRS web site page where you can start the process.