FeedController.cs

public class FeedController : Controller
{
    // GET: Feed
    public ActionResult Index()
    {
        // Webサイト上の PackageBin フォルダ以下にある vsix ファイル一覧を取得
        var entry = Directory.EnumerateFiles(Server.MapPath("./PackageBin"), "*.vsix")
            .Select(Parse)
            .Where(e => e != null)
            .ToArray();
        var feed = new feed
        {
            entry = entry,
            //ここのIDは適当に
            id = "uuid:bcsdsdd5-9999-9999-9999-7d9e16652433;id=1",
            title = new feedTitle {type = "text"},
            updated = DateTime.Now
        };
        return new XmlResult(feed);
    }

    private feedEntry Parse(string path)
    {
        //VSIXはZIPで圧縮してあるので解凍して中身を触る
        using (var archive = ZipFile.OpenRead(path))
        {
            //ほしいのはzipの中にある extension.manifest というファイル
            var entry = archive.Entries.FirstOrDefault(f => f.Name == "extension.vsixmanifest");
            if (entry == null)
                return null;
            // XmlSerializer を使って中身のXMLをデシリアライズ
            var xs = new XmlSerializer(typeof(PackageManifest));
            using (var stream = entry.Open())
            {
                var manifest = (PackageManifest)xs.Deserialize(stream);
                //VSIX1つに対応するAtomフィードのfeedEntryクラスを作成
                return new feedEntry
                {
                    author = new feedEntryAuthor
                    {
                        name = manifest.Metadata.Identity.Publisher
                    },
                    //カテゴリはVSIXの外で管理するけど、面倒なので決め打ち
                    category = new feedEntryCategory
                    {
                        term = "roslyn"
                    },
                    //このVSIXのDL先。サーバー上のファイルをそのままDLしてもらう。
                    content = new feedEntryContent
                    {
                        src = $"PackageBin/{Path.GetFileName(path)}",
                        type = "application/octet-stream"
                    },
                    id = manifest.Metadata.Identity.Id,
                    //作成・更新時刻はファイルの更新日時で代用
                    published = entry.LastWriteTime.LocalDateTime,
                    summary = new feedEntrySummary
                    {
                        type = "text",
                        Value = manifest.Metadata.Description
                    },
                    title = new feedEntryTitle
                    {
                        type = "text",
                        Value = manifest.Metadata.DisplayName
                    },
                    updated = entry.LastWriteTime.LocalDateTime,
                    Vsix = new Vsix
                    {
                        Id = manifest.Metadata.Identity.Id,
                        Version = manifest.Metadata.Identity.Version
                    }
                };
            }
        }
    }
}

//オブジェクトをXMLとして返すためのクラス
public class XmlResult : ActionResult
{
    private readonly object objectToSerialize;

    public XmlResult(object objectToSerialize)
    {
        this.objectToSerialize = objectToSerialize;
    }

    public object ObjectToSerialize => objectToSerialize;

    public override void ExecuteResult(ControllerContext context)
    {
        if (objectToSerialize != null)
        {
            context.HttpContext.Response.Clear();
            var xs = new XmlSerializer(objectToSerialize.GetType());
            context.HttpContext.Response.ContentType = "text/xml";
            xs.Serialize(context.HttpContext.Response.Output, objectToSerialize);
        }
    }
}