The shooters were never found. A Jewish nurse treating him before his death says that he said he was shot by Arabs (NYT 6/5/48) , and a United Press reporter also said that Wasson told him that the snipers were Arab (NYT 5/27/48):
Wikipedia says that others blamed a Jewish sniper, but the evidence is weak indeed -an American Colony resident who wrote in her diary "We heard today the U.S. Consul General Mr. Robert [sic] Wasson was shot by Jews on Friday and died today." but she took out the "we heard" in a 1960 book; also Amman diplomats blamed Israel, and they are not exactly the most reliable reporters.
The preponderance of known evidence shows that he was killed by Arab snipers.
The other American mentioned in the Palestine Post article, Herbert Walker, was killed in a separate incident from machine gun bullets:
