The use or implies singular, therefore have is incorrect. So we can eliminate A, B, C
Between option D and E:
Thom Tillis is:
1. Republic challenger of Senator
2. Speaker of the North Carolina
In option D
the Republican challenger seems correct, but use of back to back modifiers in continuation is incorrect.
Option E corrects this by using and for the 2 modifiers
In the high-profile Senate race in North Carolina, neither Senator Kay Hagan,
who is the Democratic leader, or Thom Tillis, who is her Republican challenger and Speaker of the North Carolina House, have released recent campaign fundraising figures.
A. who is the Democratic leader, or Thom Tillis, who is her Republican challenger and Speaker of the North Carolina House,
haveB. the Democratic leader, and Thom Tillis, her Republican challenger who is Speaker of the North Carolina House,
haveC. who is the Democratic leader, nor Thom Tillis, who is her Republican challenger and also Speaker of the North Carolina House,
haveD. the Democratic leader, nor Thom Tillis,
the Republican challenger, who is Speaker of the North Carolina House, has
E.
the Democratic leader, nor Thom Tillis, her Republican challenger and Speaker of the North Carolina House, has