Yes, you can apply for an Individual Taxpayer ID Number for your son at the IRS. You can visit your local IRS office or call them toll free at 1-800-TAXFORM and request a Form W-7A for Taxpayer ID Number for Pending U.S. adoptions.
Is there any way I can prove his U.S. immigration status?
To prove his status, you need a current U.S. immigration document. For example, an I-551 Lawful Permanent Resident Card or an I-766 Employment Authorization Document.