Yep, you don't need to be a citizen to hold Vanguard accounts or have a 401K or anything of that nature.
You do need a SSN, which obviously you have, as you have been living here before.
Do you or your wife, or both have a job lined up? Because if you don't that cold hard cash will be needed to buffer your landing, until you are settled. Depending on if you have a credit score, you may have to put deposits down to turn on utilities and such, so you may want to have some cash buffer to get you started. You can still decided where and how to invest, once everything is coming together after the move!
Also, I'd recommend (if viable) selling or transferring to a US account any stocks or funds you have invested in Europe, as the reporting requirements for both you and your bank, due to FACTA are a PITA and your European bank may actually force you to close your investment account because of that (happened to me)!!
I think it's great that you are planning to start real small, albeit in an expensive city. It's also good to have a support network, which you seem to have, as you are renting from family.
I believe it's extremely important to get a job (or some sort of income) lined up ASAP so you don't have to dip too much into your savings.
I have seen many Europeans give up here in the US, as they came over with high expectations of their experience and skills being recognized and ended up doing odd jobs instead just to keep afloat.
One of my friends will be going back next month, because she could not make it work and realized she was better of in her home country.
That said, as you have lived here before, you are probably very aware of all that.
There's something exciting about every new beginning!! I wish you and your family the best of luck!!