My Engine mounts keep braking.

Similar issue and solution on link
https://community.cartalk.com/t/96-dodge-neon-stalling-while-idling/28632
If you have check engine lite check for code. Most parts houses where I live will check them for free.
https://westpacparts.com/96-dodge-neon-20l-sohc-idles-rough-stalls-consistently-when-stopping
https://www.youtube.com/watch?v=vFhoNykgQtc

2003 neon shaking after changing both right motor mounts ...

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With the parking brake set and transmission in drive accelerate lightly and see if the engine raises up do the same in reverse if the engine moves up an engine or transmission mount is broken

Without trouble codes you are fishing in the dark. If the engine has a cylinder that is dead at idle from a fuel injector or ignition problem, it should have a code for a miss on that cylinder. A mechanical problem like a burned valve could cause the problem without setting a code. Have you checked compression on all 4 cylinders ? ?

Really : !!

Get under the car & change all engine & transaxle mounts at on time

If you can't find & identify them-- then you not up to changing them

Have a repair shop or trans shop do it

is ther a check engine light

Have somebody stand in front of van with the hood open and watch the motor while you start the van.Put the emergency brake on and your foot on the brake,shift into D keeping your foot on the brake with your other foot give the engine a little gas and watch the motor.If a motor mount is broke the motor will lift up a little on the broken side.Be careful!

Is your car a manual or automatic? If it is a manual, did you just change your clutch? I have a 2003 Dodge Neon and when I replaced the clutch the entire car shook; come to find out the clutch was not assembled right and wasn't balanced. I installed another clutch and it is fine now.

Vibration at idle brings to mind a few suspects:

transmission mount(s) broken/missing
transmission internally worn
motor mount(s) broken/missing

(1) transmission internally worn or transmission mount broken/missing - test this by shifting into neutral at the next stop light. If the vibration disappears when you disengage the transmission, then the transmission is the source of the vibration. You'll have to inspect the transmission mounts - if they're ok, then the transmission is internally worn and can no longer smoothly "slip" while idling in drive.
(2) motor mount broken/missing - also test this by shifting into neutral at the next stop light. If the vibration persists, then the transmission is exonerated, and the vibration is coming from (under) the engine. The rubber part of motor mounts commonly crack/break and or pop out and are lost.

A photo of the front motor mount for the 2003 Dodge Neon:

probable vacuum leak query brake booster