How's the job hunt coming?
is short for
How's the job hunt coming along?
and means the same as
How's the job hunt going ?
How goes the job hunt ?
going on would not be correct since one
goes job hunting
but does not
go on job hunting
The idiom going on can be used to ask the current happening status of an activity
What's going on?
What's going on with the job hunting?
Anything going on with the job hunting?
getting on can be used in the same way, meaning how far along
How's your job hunting getting on?
How are you getting on with the job hunting?