Long story short, if you are a nonresident alien (i.e., not a US citizen and not living in the US), you are subject to US Federal estate tax (the US name for inheritance taxes) only if your US "situs" assets exceed $60,000. Shares issued by a US corporation (including "regulated investment companies" like Vanguard's) would probably constitute US situs assets.
Since your wife is a US citizen, if in your will you leave the shares to her, your estate will have unlimited "marital deduction" which will essentially eliminate any estate tax.
If your country has an estate tax treaty with the US, some more lenient rules may be applicable.
Some people (usually with significant assets) make all US investments through a non-US corporation which generally avoids the issue.
And, totally unrelated, your wife should get an SSN....