sample.aspx

function file_upload(files, index = 0) {
    var file = files[index];
    var data = new FormData();
    data.append(file.name, file);

    $.ajax({
        url: '<%= VirtualPathUtility.ToAbsolute("~/api/FileUpload") %>',
        type: 'POST',
        data: data,
        cache: false,
        contentType: false,
        processData: false,
        dataType: 'html',
    })
    .catch((...args) => {
    })
    .then(() => {
        var has_next = ((files.length - 1) != index);
        if (has_next) {
            file_upload(files, (index + 1));
        } else {
            window.location.reload();
        }
    });
}

$(document).on('click', '#upload_button', function () {
    var files = $('#upload_files').get(0).files;
    if (files.length > 0) {
        file_upload(files);
    }
}

FileUploadController.cs

private FFMpegConverter FFMpeg = new FFMpegConverter();

/// <summary>
/// 動画のアップロードとサムネイルの作成
/// </summary>
/// <returns></returns>
public HttpResponseMessage Post()
{
    foreach (string name in HttpContext.Current.Request.Files)
    {
        string pathVideo = $"{VideoFolder}{name}";
        string pathThumb = $"{ThumbFolder}{Path.GetFileNameWithoutExtension(name)}.jpg";

        // 動画の保存
        HttpContext.Current.Request.Files[name].SaveAs(pathVideo);

        // サムネイル作成
        FFMpeg.GetVideoThumbnail(pathVideo, pathThumb);
    }

    return HttpContext.Current.Request.Files.Count > 0 ? Request.CreateResponse(HttpStatusCode.Created) : Request.CreateResponse(HttpStatusCode.BadRequest);
}