Check to see if your getting voltage to it when you press in key as it may be a bad keyswitch, also check ground wire. If it is bad it is part number 397909 and is not cheap.You can order it from boats dot net...
Yes, definitely find out what's going on before dropping money. It's just as likely to be a bad keyswitch or a wiring problem as it is a bad primer solenoid.
A little off the wall I know, but I have seen some salt water mtrs where the plunger in the primer solenoid would freeze up and not open the primer valves. Just a thought.