There's a public Airtable that I want to access via API, but I don't have admin permissions to the table (just the link to it). Is it possible to access this data? If it helps answer the question, what I am trying to do is create a program that allows the user to paste the airtable link and convert that data into a .ICS file.
No, you need an admin to grant you permission. After that, you can either use scripting or Make to automate & integrate wth Airtable's API. Make is the best low-code/no-code way of doing this, without needing to know any programming code at all.