How do I deploy this app for my job: EC2, Elastic Beanstalk, something else entirely?
Are you trying to pull in live data from Trello or from your companies own internal project management tool?
An EC2 might be useful, but honestly, it may be completely unnecessary if your company has its own servers. An EC2 is basically just a collection of rental computers to help with scaling. I have never used beanstalk so my input would be useless there. 
From what I can assume from the question, you could have a python script running to pull from the API and make the changes without an EC2.
I'm tasked with creating a web app (I think?) for my job that will tracker something in our system. It'll be an internal tool that staff uses to keep track of the status of one of the things we do. It should look like trello, with cards that drag from step to step. That frontend exists, but my job is to make the system update when the cards are dragged. This requires using an API in Python and isn't that complicated to grab from/update. I have no idea how to put all of this together. My job is almost completely nontechnical and there's no one internally who knows what I'm doing except for me. I'm in so over my head here and have no idea where to begin. Is this something I should deploy on Elastic Beanstalk? EC2? How do I tie this together and put it somewhere?