My brokerage gives me a few commission free trades per month. Non-citizen, non-permanent residents cannot open an account. Are there any brokerages that offer commission-free trades and are open to people that don't have a green card?
Generally, US banks/brokerages treat US tax residents the same as they treat US citizens/permanent residents, for deposit/investment accounts purposes. You may be not a permanent resident (green card holder), but you may definitely open accounts as resident if you're a tax resident.
Usually, the distinction is in the form of your identification: as a non-resident you use form W8-BEN, as a resident (even if you don't have a green card, but are in H1B or L1 status, for example) you use W9. That is all they generally care about.
I have yet to see a bank that doesn't open accounts to non-citizens and non-permanent residents, but I have seen banks that do not open accounts to non-residents for tax purposes. I think that is the source of your confusion.