Like Philip Kendall said, in athletics, a person who throws a javelin is called a "javelin thrower". Collins Dictionary has an entry for "javelin thrower":
(athletics) a person who throws a javelin
There is the word "javelinier" (or "javelineer"), but apparently that is only used for non-athletic purposes. Wikitionary defines "javelinier" as:
A soldier who throws a javelin
There used to be the word "jaculator", which is defined in The Century dictionary and cyclopedia (published in 1906) as "one who throws (a javelin)":
There is even an entry in dictionary.com which has the same definition for "jaculator", but I'm fairly certain that word isn't used...