I’m a nonresident with an ITIN and no SSN, but I want to start saving for retirement in the U.S. Can I open an IRA account using my ITIN, or is an SSN mandatory?

As my old (now semi-retired) finance professor liked to say: “Only if you’re stupid!”
Let me explain. An IRA is a tax-deferred account. The increase in value, the dividends, the interest, etc. generate little to no income tax until you take a distribution from it. At that point, the income is all ordinary income. If you don’t have U.S. taxable income, then there are no savings from opening the account, but when you withdraw money, it will be taxable income sourced to the U.S. The amount of tax you will pay will depend on your country of residence and the treaty between that country and the U.S.
The U.S. has preferential tax rates on long-term capital gains (assets held for more than one year) and qualified dividends. An IRA distribution does not get any of this treatment. Simple example, if you have $20,000 in a U.S. IRA and it earns $1,000 of qualified dividends a year for five years and the value of the $20,000 increases to $25,000. Now you have $30,000. If you have the money in an IRA and withdraw $30,000 it will be taxed at whatever your marginal tax rate is. If you had a regular investment account, you would pay at most $1,000 in capital gains taxes and the tax for each of the five years of dividends would also total $1,000.
The math changes if you have a Roth IRA but that type of account still requires that you have earned income of at least the amount you contribute to the IRA for the year.
Then there’s your other problem. You don’t understand what IRA means. It means Individual Retirement Account. That’s individual as in one person. There’s no “we” in it.