I wrote a check to a consultant for work in 2009, and now she does not want to provide her SSN for me so that I can issue her a 1099. She has asked if I can issue the 1099 to her company instead. I wrote the check to her personally, so I don't know how to do this, if it can even be done?

You should have asked her to file W-9 before making the payment.
Talk to her again. Then you can call IRS, there is penalty for not providing W-9.

I agree, and it sounds like you also agree that that I cannot just issue a 1099 to her company. I will keep pushing for the SSN. Thanks