Charlie Broadway, 23, "poses a risk to the public and should not be approached", police said.
Detectives said two men were shot at a property near Ilton, just before 20:00 BST on Thursday.
One of the men died at the scene.  The second suffered "non life-threatening" injuries.
A 28-year-old man and a woman have been arrested in connection with the shooting but Mr Broadway remains at large.
Officers said they believed the "serious incident involving a firearm" was of a "domestic nature".
The shooting took place at Chubbards Cross caravan site.
Ch Supt Jon Reilly, from Avon and Somerset Police, said: "We're using all necessary resources to trace 23-year-old Charlie Broadway in connection with these offences.
"Due to the gravity of what happened, Charlie Broadway is potentially armed and as a result poses a risk to the public.
"Obviously we have a number of armed officers on the ground and we're using other specialist resources such as the helicopter in addition to a large team of detectives.
"But obviously a firearm has been used and we wouldn't want the public to approach Charlie."
Force incident manager Rob Pearson added: "It is a very tricky and dangerous situation. There are lots of police officers working now to try and catch him."
A man who may be armed with a gun is wanted by police after a shooting in Somerset left one man dead and another injured.