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I don't know about faxing it but, when I did it online it took about 3 days start to finish.
Usually when they ask for one for a CLI they send a letter with a 2016XXXXXXXUSD number on it and direct you to www.americanexpress.com/verifyincome and then you go through the online submission process. If you faxed it over then I would call them and inquire the status and/or ask for the USD reference number to do it online if they don't have it yet.
800-535-7790 // looks like they changed the link to https://online.americanexpress.com/ecomm/esign/en_US/verifyIncome
If the letter has the USD, card #, and your zip code you should be able to access it. Otherwise call them and get the info you need to do it.
Call Them and see if they got the form. If they didn't then ask them how to do the electronic income verification online.
A little more Googling produced this:
No card # needed to upload the form.
https://www.irs.gov/pub/irs-pdf/f4506t.pdf - just fill in the form and upload via above link
I would still call them though to check the fax status before submitting another one.