Can I connect to a Amazon S3 bucket using Power Query?

Sure. This is how I do it:

1. Create a new Lambda Function ... it can be empty for now
2. Set up a new API in API Gateway
3. Create a new GET method
3.1 Select Lambda Function for the integration type
3.2 Select the Use Lambda Proxy integration option
3.3 Select the region and type in the name of the lambda function you created in step 1
4. Edit your lambda function.
4.1 Using the AWS SDK, generate a url w/ pre-signed key for your file
4.2 Return a 303 redirect to the url from step 4.1

Here is a sample of a lambda function in python 2.7:

bucket = 'bucket-name'
key = 'path-to-file'

client = boto3.client('s3')
link = client.generate_presigned_url(
        'get_object',
        {'Bucket': bucket, 'Key': key},
        7200, 'GET')

return {
    "statusCode": 303,
    "headers": {'Location': link}
}

You can use this PowerBI query as a starting point:

let
    // Replace the URI with the "invoke url" of your API Gateway endpoint
    // See: https://docs.aws.amazon.com/apigateway/latest/developerguide/how-to-call-api.html#how-to-call-api-console
    uri = "https://your-web-api.execute-api.us-east-1.amazonaws.com/stage/endpoint",

    // PowerBI should understand the 303 redirect and follow it to your presigned s3 url
    // Make sure to set IsRetry to true so that PowerBI gets a fresh url with a 
    //   fresh presigned key every time
    raw = Web.Contents(uri, [IsRetry=true]), 

    // My file is a gzipped utf-8 tsv, so I need to decompress it
    // and tell the csv parser the delimiter and encoding
    binary = Binary.Decompress(raw, Compression.GZip),
    csv = Csv.Document(binary, [
        Delimiter="#(tab)",
        extraValues=ExtraValues.Ignore,
        Encoding=65001]) // 65001 is utf8
in
    csv

Once you get everything working with a single file, you can parameterize your API and Lambda function to accept an s3 path so that you can pull in anything from your S3 account.