My employer has given me a task to implement a new feature in their architecture, which works all the way from the api endpoints to the database [...]
Good for you. This looks like a nice internship. You get to work on the actual product, like everyone else in the team. Would you prefer an internship where you get menial tasks to do? I bet not.
This is a great opportunity to learn things and show your employer what you can do. Which I think is why you were given the work that you were. My assumption is that they want to see what you can do.
For a company, an internship is a great way to see what potential employees can do without actually hiring them. If you do well, you might get a job offer at the end. Even if there is no open position at the end waiting for you, they will keep you in your database and you will be the first to call when a new job position gets opened.
Don't take this the wrong way, but they probably don't have great expectations from you. You are an intern afterall, they know what they are getting themselves into (at least I hope :D).
So first of all, keep calm. If you stress yourself you will make silly mistakes.
Second, make a plan. It doesn't need to be very detailed, just to figure out where you need to make your changes and what that might involve. Then take this plan to whomever is in charge of you or your work and run it by them. Tell them this is how you want to proceed and ask them if they see something missing or if they have some feedback. If all good, then go to work. If you discover new things that might affect the plan just adapt it as you get more information and insight into the product.
I would suggest you build this incrementally and iteratively. Build something very basic from end to end and see if it works. Then do another pass and add even more things working, still from end to end. Then do another pass and add some more. This way you avoid spending a lot of time on one piece of it, maybe with some assumptions in mind, and when you need to build the next part you notice that your assumptions are wrong. When you iterate and increment you are building a skeleton that works, then you add some meat on it, then some more meat, then... you get the idea.
Don't rush. You can work overtime if you want to, but don't over do it, it leads to burnout, to becoming tired, inattentive and then making mistakes. There might be no stated deadline, but if you impose yourself one, then you will rush, take shortcuts, and again, to making mistakes. You will have plenty of deadlines once you get a job as a software developer, trust me :D. So for now, just do your best. That's probably all that's expected from you.