After we called Vanguard, they mailed the paperwork in the child's name with the parent as the custodian. (you provide the SSN of the parent and the child when opening the account). I think it took about a week to get the forms in the mail. You fill out the forms, and you can either provide ACH information to transfer the minimum amount in, or you can attach a personal check and send with the forms back to Vanguard. About 7-10 days later they send a letter with instructions on how to access the account and set up password, if I recall. You can do ongoing contributions if you wish to set them up, same as any Vanguard account.
Keep in mind, that you need to have earned income to open a Roth IRA so we waited until our sons worked in high school. Since they were eligible for a full refund of Federal and State income taxes withhled back (they made under the annual limit at the time), we were filing a tax return for each of them anyway to get those withholdings back. Vanguard does send the appropriate tax form to document the Roth IRA contributions.
You can't open a Roth if there is no earned income for the child. Just wanted to make sure that was clear. But once you decide to open it (when they are working) you can fund the account with other funds if that appeals to you and you are able to (up to the amount they earned, of course or the max of $5,500).
We shared a post about it here: Roth IRAs for Teens and Young Adults
if you want to check it out.