Isolation around feedline that penetrates floor or walls;
How does the motor/pump sound in the machine room?
Isolation at the piston to car? Isolation coupling in the feedline and a muffler?
Jammed feedline connection transmitting that harmonic pump vibe?

Is the ride spongy at all? Bouncing at slowdown points or final stops?

I'm sorry for not responding, I was sent to a different location.

But I'm back and we are currently doing an inspection on said elevator.

The inspector noted the vibration problem, it has to be fixed.

Everybody, thank you for your responses, they will all be taken into consideration while we try to fix the problem.

I will keep you updated. Thanks again

Here are some more details:
Jack is not telescopic
Air was drained
Line is not isolated
Motor in machine room sounds perfectly fine
Isolation coupling yes, muffler no
Ride is not spongy/bouncy
I checked the belt, everything looks perfect
We did low oil test, no change

We opened the drop ceiling and checked the pipe for vibration, there was none. The problem isn't with the length of line.

Seems you’ve eliminated several possibilities, and NO PIPE VIBRATION. Did you check for piston to car isolation? Could it be the jack packing, or is down direction smooth in all transitions?
Poster “lowoil” suggested a clog at the pump intake...all ok there?

Here's an update.

There is piston to car isolation, it's unlikely that the problem is with the jack.

No vibration from motor room and it's a new installation, clog doesn't seem likely.

The mufflers are not the air up kind.

Somebody from my shop who specializes in hydros came to take a look and the solution was to install a second muffler. But an MTA inspector came and shut that down, he said the long island railroad doesnt want any extra parts on a new build.

I'll keep you posted
Thanks again for the replies

I’m anxious to hear the fix on this one.