Both are correct. They have very, very slightly different connotations. First, if the second "the" was left out:
The Big E and Topsfield Fair are coming.
we should realize that the "the" at the beginning of the sentence says that there might be an implied "the" before "Topsfield Fair" anyways. So even if it should definitely be "the Topsfield Fair" and not simply "Topsfield Fair," the omission of "the" would still be correct.
To avoid this complication, let's consider a modified version of the sentence.
"The Topsfield Fair is coming."
"Topsfield Fair is coming."
"I am going to Topsfield Fair."
"I am going to the Topsfield Fair."
Adding "the" implies the connotation that it is the premiere fair in some sense. If you say "I am going to the fair" or "I am going to the store" this implies that there is an obvious particular fair or store that you would go to, despite the possibility of other fairs or stores existing.
If we stop here, thinking of "Topsfield Fair" as just a name for a fair, omitting the "the" seems more correct. But in this case, Topsfield is a specific location. Instead of thinking of it as a name, we can also think of it as a description: "the fair of Topsfield."
There are fairs in many towns, and the Topsfield Fair is the fair in Topsfield. Here, we use "the" because Topsfield Fair acts almost as a stand in for "the fair of Topsfield," and it would be incorrect to say "I am going to fair of Topsfield."
So, both are correct. I believe omitting the "the" and just saying "Topsfield Fair" is very slightly preferable because it is a name; "Fair" is capitalized. This distances it from the connotation of it being a fair of Topsfield.
Let's see how it's actually used. If the word "the" was omitted exactly as often as it was included, we would see "Topsfield Fair" used exactly twice as much, since "the Topsfield Fair" contains the words "Topsfields Fair." The graph shows us that "Topsfield Fair" is used a bit more than double "the Topsfield Fair," indicating that while both are commonly used, omitting the "the" is slightly more common.