Is the shift solenoid for the 2002 Toyota Camary with a 4 cylinder motor internal or external?

internal

some are external and few are internel

Get a readout from the car's computer to see if there is a trouble code stored, which will give a good clue to the problem.

Camrys of this age often have a shift solenoid fail in the transmission, which will cause the symptoms you describe,

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P1765 TOYOTA - Linear Shift Solenoid Circuit Fault

This code doesn't mean anything is wrong with the shift solenoid, it just means a problem in the circuit or a shaft not sensed. You need the dealer to properly diagnose this problem.

Your 2nd to 4th gear cluches are out. You need a Trans.

pull the transmission pan and remove the filter, you'll find the shift solenoids there.

Trans model is the 4L60E,and the shift solenoids are mounted on the valve body,just inside the bottom pan.There are 2 shift solenoids,one EPC solenoid (Electronic Pressure Control)One TCC PWM solenoid (Torque Convertor Clutch,Pulse Width Modulated),one 3-2 kickdown solenoid,and a manifold pressure switch.As far as a shift solenoid relay switch,i have never heard of that.The power for the transmission solenoids comes through the ignition switch ,and uses a fuse,not a relay.Accessing the solenoids only requires dropping the bottom pan,and the valve body does not have to be removed,to change the solenoids.The shift solenoids are the TWO solenoids right side by side and look identical..They are the only two solenoids inside the pan that are in fact identical.Good luck.

solenoids are worn out, it will probably eventually start after 4 or 5 clicks but its annoying as hell