My son’s 1999 Lexus LS400 will start shaking after he drives it for about 10 minutes… it has about 200,000 miles on it.
If it shakes when setting still or in park it would most likely be related to the engine. Probably an engine misfire. If this happens often enough the check engine light will illuminate. If the light has already come on, pull the engine codes and post them below. Most auto parts stores will perform this task at no cost.
If the shaking only occurs while moving it would be in the drive-train.