Yes, you can claim her and she can have a tax id. In Nonresident Alien Spouse | Internal Revenue Service it explains how to file a return, the elections required and their implications. It also includes:
Social Security Number
If your spouse is neither a U.S. citizen nor a U.S. resident within the meaning of IRC section 7701(b)(1)(A) and you file a joint or separate return, your spouse must have either a Social Security Number (SSN) or an Individual Taxpayer Identification Number (ITIN). To get an SSN for your spouse, apply at a social security office or U.S. consulate. You must complete Form SS-5. You must also provide original or certified copies of documents to verify your spouse's age, identity, and citizenship. If your spouse is not eligible to get an SSN, they can file Form W-7 with the IRS to apply for an ITIN. Refer to Taxpayer Identification Numbers (TIN) for more information.
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