As the description (located at http://developer.github.com/v3/repos/contents/) says:

/repos/:owner/:repo/contents/:path

An ajax code will be:

$.ajax({
    url: readme_uri,
    dataType: 'jsonp',
    success: function(results)
    {
        var content = results.data.content;
    });

Replace the readme_uri by the proper /repos/:owner/:repo/contents/:path.

I cross posted the question on the GitHub community after I quickly got a -1 vote for some unknown reason here. For anyone who has the same issue, here is a useful answer: https://github.com/orgs/community/discussions/151261 (and I suggest using the community, as the response was quick and impressive)

The following is a copy & paste of that answer:
-- start copy & paste --

Fetching all repositories a user has contributed to within a large organization in a single API call is quite challenging due to GitHub's current API limitations. However, here are some approaches that might help:

Using GraphQL API (Efficient Querying)

GitHub's GraphQL API provides a way to query user contributions without looping over repositories manually.

Using contributionsCollection to Fetch Contributed Repos The following query retrieves repositories where a user has made commit contributions:

{
  user(login: "YourGitHubUsername") {
    contributionsCollection {
      commitContributionsByRepository(maxRepositories: 100) {
        repository {
          name
          owner {
            login
          }
        }
      }
    }
  }
}

🔹 To include pull request contributions, extend the query:

{
  user(login: "YourGitHubUsername") {
    contributionsCollection {
      pullRequestContributionsByRepository(maxRepositories: 100) {
        repository {
          name
          owner {
            login
          }
        }
      }
    }
  }
}

Using GitHub Search API (REST Alternative)

The Search API supports finding repositories where a user is a contributor:

GET https://api.github.com/search/repositories?q=user:YourGitHubUsername

Consider OctoKit for Optimization

Using OctoKit can streamline GraphQL requests with better pagination handling and performance.

Conclusion

There’s no single API call that directly fetches all contributed repositories across an organization, but the GraphQL contributionsCollection is the closest built-in solution. If you need a broader dataset, batching API calls and caching results daily may still be required.

-- end of copy & paste --

In addition, on the company slack channel someone suggested looking into viewer.topRepositories, as the documentation for it says

Repositories the user has contributed to, ordered by contribution rank, plus repositories the user has created.

We didn't get around to testing this as we ended up using the GraphQL query above.