You probably have a misfire; Have it checked for codes. Most likely you need a tune up.
do u mean that does yor engine move side to side during acceleration. well no because your foot is on the accelerate
You can find a good PT Cruiser Engine online or through your a PT Cruiser dealership.
During acceleration? I'm not sure. But when both idling and acceleration the engine's combustion can cause a lot of vibration in the steering wheel if you are in an older vehicle.
Possible exhaust leak
The 2008 Toyota FJ-Cruiser has a V6 engine.
The 2013 Toyota FJ-Cruiser has a V6 engine.
The 2010 Toyota FJ-Cruiser has a V6 engine.
The 2008 Toyota Land-Cruiser has a V8 engine.
The 2006 Toyota Land-Cruiser has a V8 engine.
The 2011 Toyota FJ-Cruiser has a V6 engine.
The 2013 Toyota Land-Cruiser has a V8 engine.
The 2014 Toyota FJ-Cruiser has a V6 engine.