You haven't said what make and model of car it is.
On a rear wheel drive car a worn universal joint in the driveshaft (propshaft) can cause the symptoms you describe. It's sometimes referred to as a 'yoke'.
Wear causes the universal joint/propshaft to be out of line/balance and causes the vibration when the car is moving. Another symptom is a 'clunk' when the car engages gear. That clunk noise is also caused by wear.
You can check it quite easily. get underneath the car, grip the propshaft and try and turn it clockwise/anti clockwise. If there's any movement - it's worn.
The link below is of generic images
propshaft universal joint Google Search
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