I like to use this web site for quotes. It takes very little information to get started. Great rates too! Good Luck!

Both answers are correct, but to add to that, if you are trying to compare two quotes to see what would be a better fit and then once you have chosen and they run your credit and find out something is off the only one you are hurting in all of this is yourself. You are going to be disappointed if something comes up and you have already made up your mind you are going to be very disappointed. If you are serious about buying you need to be up front and factual about everything so you are given the best service possible.

Actually, if you want someone to give you a GFE that holds water, you should be prepared to give factual information that pertains to your loan this includes running a credit report. You are not penalized for multiple inquiries within a 15 day period within the same industry.
If you do not give enough information, be prepared to not receive an accurate quote.

Yes, absolutely. Keep in mind that when a mortgage professional is providing a good faith estimate and they do not have your credit report or full loan application, they are basing it on what you tell them. There have been many times a client I was working with told me they had a 750 FICO score and so I based the GFE on that, yet when I pulled their credit it more like 710. There is a big difference in rates/pricing with those scores.

So to answer your question, you shouldn't have to give your SS#, but they will be basing the quote on the information you provide.