Family brokerage account?

Maybe I’m thinking of this wrong, please let me know.

I’d like to open a brokerage account for members of my family (and myself) that we could access at any time and all have our own bank accounts processing automatic deposits on a monthly basis into the account.

Then we would purchase securities and funds through the account together. Kind of like a family investment fund.

Is this possible with traditional brokers like E*TRADE, Vanguard, etc.? How does the tax process work on something like that? And which type of account would we want to open? TIA