When I was faced with a similar situation about 10 years ago, Vanguard turned out to not be very friendly to small investments and starter accounts. My kids ended up opening their accounts at Schwab, and that has turned out very well for them. Schwab is friendly to very small accounts, has low costs, and in my case has a local office which has turned out handy a time or two.
In general, kids under 18 aren't allowed to open their own bank or investment accounts. But you can open an account for them and be the custodian. The money is theirs, and will be taxed under their SSN, but you'll be the one managing the money until they reach adulthood.
There are two other options:
1. You could open a separate account in your name that would be your money as far as the financial institution is concerned but mentally allocate it to them. Then later, when they are ready, you could just gift them the assets in the account; if it's under $15K per person per year there would be no taxes, and they would get your holding period and basis.
2. If you want to save for their college, you could look into opening 529 accounts for them. These usually can be established with very little to start with, and have a custodial structure to them where you are the custodian but it's their money.
