I would have to say Southwest took about a month, and called directly to ask a few questions, then the tickets came fedex. I think the hold up was that they couldn't mail to a PO box, which my organization has as a mailing address. So, the mail came directly to my house. That is the case with many donations. Companies will not always send to a PO Box. I did not write to them this year, so can't remember exactly. Here is the exact link:
southwest.sponsorport.com/ You have to live near a hub city for Southwest, and you need to upload a copy of your 501(c)(3) determination letter from the federal government. That may be a tough one if you are using a town wide number.
If you go on a website, always look for the "contact us" area. That is where you can start to send a request. If that is not the right place, they generally will e-mail you the correct place to send a request. Also, try to get to the corporate web page, rather then the customer website. You can usually find a link under "About Us" or sometimes they are helpful and actually have a link to community involvement.
For the authors, as I noted, I sent a message via the contact us tab. I started it by saying how much I loved their books, noted a particular one that I read recently and why I liked it, then added that I would love to have them donate to a fundraiser I was helping to organize. I stated that below was my letter, pasted the entire solicitation letter, then ended by saying thank you for taking the time to read the letter. I did the same for the other donors I noted, and had great success. Always start an e-mail with how much you love the product, that gets them to keep reading.