Intermittent vibration

My wife's car has recently developed an annoying vibration. It is an intermittent issue, and when it does occur it is only under acceleration. If you lift your foot off the gas it goes away. As stated it is intermittent meaning it sometimes does it and sometimes drives fine, usually only occurs if you drive above forty five mph for a prolonged amount of time. One thing I have noticed is that if you turn hard at and cause the body to roll and put weight on either side of the car it goes away for a little while. We do not really have the extra money to have it diagnosed or to buy parts we do not need. Any idea what could be causing this? It started after a breather hose broke off the top of the engine (part of EVAP system I believe.