It sounds like you're already dong User Lookup, like this, but I mention it just in case:
var users =
(from user in twitterCtx.User
where user.Type == UserType.Lookup &&
user.UserID== "123,456,789,...,777"
select user)
.ToList();
users.ForEach(user => Console.WriteLine("Name: " + user.Name));
As you know, you're limited to 100 IDs at a time, meaning that you'll need to make a lot of requests, which will take a long time. Check out Twitter's rate limiting docs, which explain how many requests you can make and (more importantly) techniques that might help to avoid lower rate limits:
https://dev.twitter.com/docs/rate-limiting
In LINQ to Twitter, you can access rate limit info through the Headers and other properties on your TwitterContext instance right after a query.
Another potential option is to look at Twitter's Partner Providers to see if they have the data you need:
https://dev.twitter.com/docs/twitter-data-providers
Joe