If I'm transferring funds to a payee (sellers, vendor, etc) via the API, do I need to issue them a 1099-MISC?
You should definitely consult the IRS guidelines or your own tax professional, since the reporting requirements are very specific to your business and contain numerous specific exceptions. But in general, a 1099-MISC needs to be issued under situations when you pay someone for rent, services, royalties or other payments made in connection with a trade or business. 1099-MISC forms must be issued if you have paid $600 or more to a non-employee individual or business.
Stripe currently only issues 1099-K forms for Stripe users to whom we provide payment processing services and who have passed over the thresholds established by the IRS (currently, $20,000 in gross volume per year and 200 or more transactions). That is, Stripe will issue you (the Stripe user) a 1099-K tax form, but it is your responsibility to file 1099-MISC forms appropriately for any of your payees.
To help you do that for tax year 2016, we are partnering with Payable to help you provide 1099s to your contractors or vendors. You can find more details here.