I finally figured this out. In order to avoid cross-site scripting issues, I created a proxy endpoint on my server. Then I can pass the image URL to my server, which then executes a GET request on the remote file, converts the response to Base64, and sends it back to the browser. The browser can then convert the data back to binary and create a Blob (which is as good as a File for my purposes).
$.ajax({
url: apiRoot + "/proxy",
data: {url: "http://example.com/image.jpg"},
success: function(data) {
var binary = atob(data.split(',')[1]);
var array = [];
for (var i = 0; i < binary.length; i++) {
array.push(binary.charCodeAt(i));
}
var file = new Blob([new Uint8Array(array)], {type: 'image/jpeg'});
}
});