Stackexchange REST API has been blocked by CORS policy

does not StackExchange API attach in the response header Access-Control-Allow-Origin? I'm doing something wrong in how I'm invoking the request?

The Stack Exchange API permits client-side requests, but the page you're requesting is not the Stack Exchange API - it's just a plain Stack Exchange page, not the API endpoint designed for such things, so it doesn't have the required header.

If you want to use the API properly, go through the documentation here and choose the endpoint you need. For an example using my user ID, one might use the following code:

fetch('https://api.stackexchange.com/2.2/users/9515207/reputation?site=stackoverflow')
  .then(res => res.json())
  .then(console.log);

api.stackexchange.com has the proper headers: access-control-allow-origin: *, which allows for client-side scripts to request it.

since this request is for a trial project, can I implement a proxy using a free-tier of some platform? I see a lot of suggestions for Heroku but I'm not very familiar with that platform

For the more general case of a server that doesn't set the required headers, if you wanted to use one of their responses client-side, you would have to bounce the request off a relay server. You could either make your own app on Heroku (a good free option), or use something like https://cors-anywhere.herokuapp.com/