I have a 89 Mazda 626 with a similar problem. Sometimes when I go to start it it won't start. (It does not click like yours does.) All I do is put the car in neutral and turn the key. It starts right up. Maybe this could help you.
Dead battery? Loose or dirty battery cables? Bad starter solenoid? Bad starter? Neutral switch?
Probably starter solenoid switch
Check the battery cables or starter.
Voltage Regulator?
Weak battery? Loose or corroded battery cables? Starter solenoid? Starter? Neutral Switch?
Dead battery? Loose or corroded battery cables? Bad starter? Bad starter solenoid? Bad neutral Switch?
Seized engine Discharged battery Loose or corroded battery cables Starter or starter solenoid Neutral safety switch
Seized engine? Loose or corroded battery cables? Weak or dead battery? Bad starter solenoid? Bad starter? Bad clutch or neutral safety switch? Bad ignition switch?
Double check the battery cables at the battery and at the engine, be sure they are clean and tight.
Weak battery Bad starter Loose or corroded battery cables Bad neutral switch
Engine seized? Dead battery? Loose or corroded battery cables? Bad starter? Bad starter solenoid? Bad neutral or clutch safety switch?
Seized engine? Weak or dead battery? Loose or corroded battery cables? Dead spot on starter? Bad starter? Bad starter solenoid? Bad neutral or clutch safety switch?