Your employer (if follow the legal rules) must comply with the goverment and fill out with you the form I-9 that is eligibility verification. There are certain criteria that you must have and none of them is an ITIN number (check the I-9 form), normally ITIN numbers are for certain people like a person who is NOT eligible for SSN which is not your case, please check:
http://www.irs.gov/individuals/article/0,,id=96287,00.html#what
Anyway, the waiting period to get your SSN will depend on your SSA office and if your information on USCIS database is updated on time.
Most of the time you need to show to your employer within a month of work, but that depend in their HR department, your EAD card and your SSN card.
Anyway you need your SSN to pay the proper taxes when you received your salary and not only the federal taxes the state taxes as well.
Check with your local SSA office what are the documents that you should bring as a must, even the norm is the one that appears at ssa.gov, you can download the pdf form S-5 from the website. Normally everything should be in original and includes your passport and the EAD card or other document for identification, but since every SSA office apparently has its own rules even they all should follow the federal, it's better to ask before you arrive and wait. There is a person that the office ask for the I-94, so as I say that depends...I never had any problem and did not have to wait more than 7 days, but there are many people who had to wait over 4 weeks. Actually the day after, I got my number so I could get my DL right away and 5 days later the card arrived on the mail.
Good luck,