I was a full time student during the Spring and Summer of 2009. I received the initial bill for the Spring 2009 Semester in December of 2008. I received subsequent statements for scheduled payment during the Spring of 2009. All payments were made in the Spring of 2009. I have all of the statements and proof of payments. However, my 2009 1098-T does not reflect those payments because, I suspect, the initial statement was prepared in December 2008. Is there anyway I can claim those payment in 2009?

If you can prove that you made the payments in 2009, then claim them on your 2009 tax return.

For 1098-T is the guide from which the IRS will monitor your claim for education credits, so they may ask for proof of when you made the payments. However, if you can prove you made those payment in 2009, then 2009 is the year you should claim the deductions/credits.