I, as an individual, paid over $8,000 to handlers to have my dog AKC Championed. Can I issue them a 1099-Misc form?

Yes you can, but only if the payments were in connection with your trade or business. You do not use 1099-MISC for personal payments to individuals. Have you been withholding federal and state (if applicable) income tax from their pay during the year?

For instructions on 1099-MISC, see: http://www.irs.gov/pub/irs-pdf/i1099msc.pdf