Figured it out. A combination of react-file-reader and HTML5’s FileReader (see this page).
Placed the react-file-reader bit inside of render:
<ReactFileReader handleFiles={this.handleFiles} fileTypes={'.csv'}>
<button className="btn">Upload</button>
</ReactFileReader>
And then this above.
handleFiles = files => {
var reader = new FileReader();
reader.onload = function(e) {
// Use reader.result
alert(reader.result)
}
reader.readAsText(files[0]);
}