No one knows who the first person was who added mustard to a hot dog. The hot dog & mustard combination was popular by at least 1851, when Charles Dickens wrote about it.
     
     
Google Books
Household Words
A Weekly Journal
conducted by Charles Dickens
Volume III
From March 29, 1851, to September 20, 1851.
London: Office, 16, Wellington Street North
1851
“The Key of the Street”
Pg. 567:
Here are crowds of customers, hot and hungry from the Lyceum or Drury Lane, and clamorous for sandwiches. Ham sandwiches, beef sandwiches, German sausage sandwiches—legions of sandwiches are cut and consumed. The cry is “mustard,” and anon the coppers rattle, and payment is tendered and change given.
   
24 October 1884, Grand Forks (ND) Daily Herald, pg. 4:
It was on a Coney Island boat. (...) The big elephant was blurred from his sight, and the gleaming houses on the island lost their interest for him,and he sighed when at last he saw the heartless damsel masticating a toothsome Frankfort sausage between two crusty pieces of bread and taking her place on a merry-go-round.
 
18 January 1890, Salem (OH) Daily News, pg. 5, col. 1:
29 January 1890, Knoxville (TN) Journal, pg. 7:
CHICAGO’S NIGHT COOKS
[From the Chicago News—ed.]
(...)
“This class is the most common,” said the detective. “See, he sells hash, bread and Frankfort sausage, red-hot.”
 
“Vill de shentlemens haf some red-hots und brod?” asked the cook, as he placed his copper kettle on the curb. In a twinkling the table was set up. His wares were good. Hot, home-made hash, with good bread and butter, made excellent sandwiches for a hungry rounder or policeman. The red-hots were generally cut in two longitudinally and smothered in mustard. The merchant willingly told how he made his living.
   
“You see, frents, I sleeps me in de day-time, ‘cause de beeblers what vants mine stock dey be sleepin, too. Mine woman, she cooks de hash ofery afternoon und I cook de red-hots vile I carries dem. Lots of fellows make money mit dis business. See, in dis part I keeps de hash, and here are de red-hots. Under is de lamp what keeps de blace hot. In dis box I carries the brod and mustard. I shust valk me round, und de peoples what is hungry dey buys. Dey be beoples vhat only work aroun’ nights. Some be tieves, some gamblers, some policemen and odder ting. Oh yes, I make more money als vorkin’ in a restaurant.”
 
8 November 1890, Lowell (MA) Sun, pg. 2, col. 1:
8 November 1890, Grand Forks (ND) Daily Herald, pg. 3:
Cheap Wall Street Lunches.
With the fall weather the Hamburger sausage has made its appearance in Wall street. The junior clerks and messenger boys who work in that section of the city patronize street lunches extensively. In summer fruit, cake and sandwiches seem to be very popular, but with the cooler temperature the sausages attain great vogue. They are dispensed from steamers in which they are kept hot, and are served in these long, narrow rolls, with, if the purchaser desires, a dash of mustard. And really they are very good. I have noticed that they are bought by men who evidently are not forced to get them on the score of economy, and the number carried away by office boys when they have finished their own is somewhat striking.—New York Telegram.
 
23 September 1894, Duluth (MN) News-Tribune, “Chicago at Night,” pg. 3:
More numerous than the lunch wagon is the strolling salesman of “red hots.” This individual clothed in ragged trousers, a white coat and cook’s cap, and unlimited cheek, obstructs the night prowler at every corner. He carries a tank in which are swimming and sizzling hundreds of Frankforters or Wieners. These mysterious denizens of the steaming deep are sold for five cents, which modest charge includes an allowance of horseradish or some other tear-producing substance.