I have been studying the documentation for the Dropbox API but I couldn't find a way to directly access an account without going to the OAuth process. Is there a way to achieve that?
My final goal is to have a webpage with a list of files and folders from a specific Dropbox account (my own), which can be viewed and downloaded by anyone.
To access a user's Dropbox account via the API, your app will need to be authorized by the user. The Dropbox API currently requires that this authorization be done via the OAuth flow. You only need to perform this step once per user though, as you can store and reuse the access token for each user.
It sounds like you intend to use only one account though (your own), so you can just process this flow once manually yourself, and save and reuse the access token programmatically.