How can an Israeli resident invest in a US brokerage account?

You don't need an US taxpayer identification number in order to invest money through an US brokerage, if you're neither a citizen nor a resident. As far as tax goes, it should be sufficient to fill out a Form W-8BEN for the broker's records.

At least, that is my experience when I needed a brokerage account when my employer wanted to issue stock in their American parent corporation to me. However, I suspect the employer may have paid the broker to accept me as a customer; the account has been conspicuously free of maintenance fees, or even fees for wire transfers. I don't know how easy it is for a complete stranger to find a US brokerage that will do business with you, but at least there doesn't seem to be any regulatory barriers to doing so.

If you're not planning to do frequent speculative trades, it may be more convenient for you to find an Israeli financial institution that can trade on the US markets for you. This may (or may not!) be more expensive for you in fees out of pocket, but can potentially save you a lot of tax headaches.