Should a nonresident US citizen get SSN

The US taxes all global income of all US citizens. It doesn't matter where you live, work, or die, or if you ever set foot in the US - if you are a US citizen, you are taxed.

That means every US citizen with income is required to file his taxes (unless he makes less than the (current=2020) limit of $ 12 200per person) - and for that you need an SSN or an ITIN.

Note that the tax liability might well be zero every year: if a US citizen pays taxes in another country for his income, and the US has a no-double-tax agreement with that country, the tax already paid is deducted from the US taxes due (which might and often does result in 0 or less).

I don't know if Israel has such an agreement with the US, but I would think they do.

Btw, you can buy shares of US companies or ETFs of them, without buying them in the US - most country's providers offer that. No need to open an account remotely in the Us.

You have asked a new question in comments on the other answer

We have also visited the US for our honeymoon so if there were any issues, wouldn't they hold my wife for a bit as soon as she was trying to enter the US?

that cannot reasonably be addressed in comments. This is against Stack policy; this is not supposed to be a chat room, questions are supposed to be questions, answers are supposed to be answers, and comments may be deleted at any time.

But this displays several (more) serious problems.

First, IRS and FinCen probably don't know your wife has failed to file. For much economic activity occurring within the US, IRS gets 'third-party' reporting -- from (as applicable) your employer(s) or retirement plan(s), your bank(s) and/or broker(s), your mortgage lender, your school, 'payment networks' like paypal, casinos and lotteries if you win, certain businesses if you deal in large amounts of cash, and so on. While they don't know everything you do, if you have a reasonably normal life they have a pretty good idea if you should be filing and don't.

OTOH IRS doesn't learn about most foreign (= non-US) activity unless (and to the extent) it is voluntarily reported. Since FATCA has been implemented in the past few years, FinCen does learn about some (much?) foreign activity by people using US identification and/or addresses, but since your wife (I presume) has Israeli identification and address the bank(s) she deals with probably don't identify her as a 'US person' even though according to the law she is one.

Second, even if IRS did identify her as a nonfiler, and even find a way to assess her tax and peg her as a nonpayer (which is what they care most about, of course), it wouldn't affect entry. Immigration and Customs are part of DHS, which is a different part of the government; they have no responsibility for tax enforcement and have enough work already that they don't look for more. (Well, not income tax; they are responsible for some import taxes.) If you owe more than about $50k tax and don't pay it, under a recent law IRS will 'certify' you to the State Department who are required not to issue you a passport (including a renewal), and permitted to revoke a passport you already have. For most US citizens, not having a valid US passport effectively blocks nearly all travel, because they can't get any other, but dual citizens like your wife can have another passport regardless of any US tax issues. (Technically, US law requires that a US citizen -- even one with dual citizenship and a passport from the other country -- must 'bear' a US passport when leaving and entering the US. Since US has no exit checks, leaving without it won't get caught. Trying to enter without it could cause trouble IF CBP identifies you; even an Israeli passport showing a US birthplace might be enough for that. There are numerous Qs about this on travel.SX; search there.)

And there's no statute of limitations for failure to file. If they catch her 30 years from now, they can assess all 30 years of taxes, and if you're even similar to normal people you won't have saved all the records needed to dispute that. Once you file, for residents absent fraud they only have 3 years to catch anything they think you did wrong; I know this is extended moderately -- I think to 5 years or so -- for some foreign items but I don't remember exactly what, or where I saw it specified; if I find it I will update. (Nit: technically ASED is 3 years from the filing due date or actual filing date whichever is later, but normally you can't file more than 2.5 months before the due date so that's a small difference. This year because COVID the window for 2019 returns is 3 months larger.)

FWIW, she does qualify to receive the 'Economic Impact Payment' of $1200 for one adult for this year (and assuming no dependent children). This was an emergency response to COVID and most people already on file with IRS or SSA/RRB/VA received payments (direct deposit or check) in April-June, but technically this was actually an advance refund of a credit on 2020 tax, so someone who didn't receive the 'advance' can get it when filing their 2020 return in early 2021.