I want to do stress/load test for the payment process of an E-commerce app like "Flipkart" or "Amazon".
Will Apache JMeter be able to handle a load of 10,000 users and handle session keys since the users will be logging in through API calls?
JMeter doesn't have limit of virtual users it can simulate, the number of virtual users you can mimic depends on several factors, to wit:
See What’s the Max Number of Users You Can Test on JMeter? article for more detailed explanation if needed.