Can we clear the bushes around our foot stones by ourselves?

I would. If you pay for perpetual care, I would take pics & bring the condition to their attention prior to doing anything.

I would call and discuss this with the cemetery because those plantings could be paid for my someone else and there could be a seasonal care contract paid for. Also, there could be a liability issue. I don't think anyone would say anything about raking some leaves but if you show up with a power trimmer, you will probably be asked questions. Their phone number is: 201-262-1100

I do not see why you cannot do that, but mention it to the appropriate people so they can bring attention to the landscaper or contractor they have on contract