You don't issue a W9 to anyone.  You give someone a W9 to fill out for them to give you their ssn or ein.
If you paid someone $600 or more from your business you might need to give them a 1099Misc.  
You can put either your EIN or SSN on the 1099Misc.

What are you trying to do?  

I recently started doing freelance design work. A client requested my SS #, which I sent them. Now they asked for my EIN which due to the govt. shutdown I cannot get as the dept. that issues them online is closed. I'm just trying to figure out what I need to do so they can issue me a 1099 form. Thanks.

If you have their EIN or SSN, then you have no need to provide them a W-9. The W-9 is sent to a person so they can provide that information to you.

They don't need both to issue you a 1099-MISC. If you've already provided your SSN, then they have all they need.

Hi, its the other way around. They have my SSN and need an EIN from me, which I cannot get.

Do you have a EIN?  They only need your SSN or EIN not both.  You don't have to get an EIN unless you pay employees on a W2.  Otherwise it is optional.  People get EIN so they don't have to give out their ssn to anyone.

Hi Carl. Thank you so much for the clarification. I'm sure the accounting dept. is aware of this but I wanted to research before I talk to them later today (about this among other things). Again thanks! Fernando

Are you unable to get an EIN through the online system at <a rel="nofollow" target="_blank" href="https://www.irs.gov/businesses/small-businesses-self-employed/apply-for-an-employer-identification-n...> ???
You should be able to do so right now, based on the "hours of operation" posted on the website. There's no mention of the shutdown affecting anything there.

I definitely will but the website is not cooperating. Thanks!

If you already gave them your ssn they don’t need an EIN from you.