If you do not want to pay for a gift subscription all at once, you can order a regular monthly recurring subscription and set the ship to address to the person you are gifting the box to but the billing information would be your information. You could also change the email address in the account to the person you are gifting to so that they would get the shipping notifications each month. If you want the person you are gifting to be able to mange the account themselves (so change their size at anytime themselves, skip months, change shipping address, etc.) you can give them access to the account by providing them with the username (which will be the email on the account) and password (that you create and they can change at anytime).
If you want a gift subscription to be separate from your own account dashboard, when you order, order under a different email address, otherwise if you get the box already, the 2nd subscription will be added and managed under your existing box account, and not as 2 separate account dashboards.
One thing to note is that the 1, 3, 6 and 12 month subscriptions do not renew after the time frame (so if you bought 3 months, after 3 months it would not renew for another 3 months). The monthly non-gift versions will renew indefinitely until it is cancelled. So if you did not want to pay for more than say 3 months, you would need to cancel after 3 months, or you would need to have the person you are gifting to change the billing info to their billing information if they want to keep it.