If you buy directly from Disney, your Guide can call Member Services to put a reservation on hold for you as soon as you give him/her your credit card number to make your deposit.
The hold lasts for 24 hours. It may take about that long for Disney to enter you into the computer. When you are entered into the computer, your Guide will call you, give you your member ID number, and patch you through to Member Services. Member Services will then book your reservation. Member Services can also give you your Club ID number (which is different from your member ID number) and your one-time activation code; you can use those two things to register on the website.
If the person you speak to at Member Services tells you that they cannot give you your Club ID number and one-time activation code, they are misinformed. Call back and speak to someone else.
All of this can happen before you get your paperwork in the mail.
But remember that Member Services is only open from 9:00 am to 5:30 pm.
If you did not buy direct from Disney, that's a whole different ball game. There are threads here about the timelines for resale purchases.