Try simply adding the following to your package.json: "type": "module"

I tested this with import * as Mastodon from 'tsl-mastodon-api'; in a test.js file using the example from their readme, executing with ts-node test.js (NOTE THAT THIS IS A JS FILE! Not a .ts file) and it works. It times out, of course, because the example URL doesn't exist, but you get the idea. Minimal example (package.json is default from npm init -y but with the "type": "module" added. I also tested with your package.json [adding type:module] and tsconfig.json):

test.js :

import * as Mastodon from 'tsl-mastodon-api';

const mastodon = new Mastodon.API({
    access_token: 'ABC',
    api_url: 'https://mastodon.example/api/v1/'
});

const result = await mastodon.postStatus({
    sensitive: true,
    spoiler_text: 'Hello',
    status: 'World'
});
console.log(JSON.stringify(result));

Full package.json that you provided with the "type" property added :

  {
    "name": "hourlypets",
    "type": "module",
    "version": "1.0.0",
    "description": "A bot for displaying hourly pets",
    "main": "dist/index.js", 
    "scripts": {
      "start": "node dist/index.js", 
      "test": "echo \"Error: no test specified\" && exit 1"
    },
    "keywords": [],
    "author": "",
    "license": "ISC",
    "devDependencies": {
      "@types/node": "^20.5.1",
      "typescript": "^5.1.6"
    },
    "dependencies": {
      "@petfinder/petfinder-js": "^1.0.6",
      "dotenv": "^16.3.1",
      "tsl-mastodon-api": "^0.4.0"
    }
  }

On the note that I used test.js - if this was instead test.ts and you tried to execute it directly with ts-node like ts-node test.ts it will throw the error TypeError [ERR_UNKNOWN_FILE_EXTENSION]: Unknown file extension ".ts" for test.ts. So, I made the assumption that your file is a JS based solely on that fact.

If you still have problems, post a super simplified script like my test.js above that causes the issue for you and I'll test with yours. You can exclude all private code - just enough to trigger the module error... meaning you could probably just have the import and console.log('hi'); and that would be sufficient.

At one point I did have if set to commonJS too, but it seemed to cause issues. Sending the one the is updated as a suggestion