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.