what should I put for my business start date?
In terms of filing for your EIN, the business start date is the day you formed your business (i.e. the day you registered your business with the state). It's not when your website goes live, or when you make your first sale. You can make an educated guess as to when those things will happen, but you can't say with 100% certainty when they will.
Should I just wait until I actually open my doors for business to get my EIN?
No. Get your EIN now. You'll need your EIN now to get certain things going for your business.
Example 1: You'll need an EIN to open a business bank account, and you'll want to open your business bank account before you launch your online business. You'll have expenses leading up to when your business goes live, and you'll want those expenses to come out of your business account, not your personal bank account.
Example 2: An online business implies a website. Depending on the nature of your business you may want to get an SSL certificate for your website. You may need to provide your EIN to the SSL certificate provider before they will issue you a certificate.
This fairly detailed document created by the IRS may be of help to you: Understanding Your EIN