Dictionary supports thread safe reading. Here is the proof from MSDN:
A Dictionary can support multiple readers concurrently, as long as the collection is not modified. Even so, enumerating through a collection is intrinsically not a thread-safe procedure. In the rare case where an enumeration contends with write accesses, the collection must be locked during the entire enumeration. To allow the collection to be accessed by multiple threads for reading and writing, you must implement your own synchronization.
So, if you are planning to only read data from it, it should work. However, I do not believe that your dictionary is filled only once and won't be modified during your application work. in this case, all other guys in this thread are correct, it is necessary to synchronize access to this dictionary and it is best to use the ConcurrentDictionary object.
Now, I want to say a couple of words about the design itself. If you want to store a shared data between users, use ASP.NET Cache instead which was designed for such purposes.