Reading your post I get chills, I'm not trying to disrespect you in any way as your most likely fully competent in your comfort zone but you shouldn't be entertaining doing any such work without the knowledge and training required, motor controls and circuits can be simple in operation yet complex in there fault finding or vice versa, getting it wrong can be very costly, you clealy are delving into areas you don't have any experience in so as an Electrical Engineer who designs builds and faults control system for machinery I hope you take my words wisely as I see the results of eager Electricians having a go at the Industrial control, I have seen companies go bankrupt from stepping into this area and making basic errors hence my response.
I'll pop a few queries and try to help here ..
Is the control circuit independent of the motor supply circuit?
Is it an E-stop or a functional stop (the shape of the button is not always what you may be led to believe).
Have you done any tests on the control circuit to establish the circuit is reacting to its button status?
Is the contactor welded in?... visual observation on this one normally.
Has the operator given any info that may help in diagnosing the problem?
If the DOL starter is a all in one unit, does the mechanics of the stop button actually press the stop on the overload as these can sometimes become warn or mis-aligned ..simple check is remove start/stop cover and press the buttons manually - this only relates to buttons that are just physical extensions and no wiring on them.