car engine keeps shaking after ignited

IMPORTANT: You will be in the engine bay -- if you have long hair, use a hair tie and a hair clip (if possible) to avoid becoming friends with a hot engine block! I've been fine with a hair tie being at shoulder length, but if it's super long you really need both - even JUST at the point where it becomes long enough to become iffy (see: 15"+) you need both!

First thing I would do is scan for codes -- that said, it's probably a misfire. Scan the car and see which cylinder is affected, and then swap a coil over first (ex: Cylinder 3 misfires, move it to Cylinder 4 and see if the code moves - if it doesn't, then the spark plugs are the next common cause of this. If you can feel it out by hand, you can potentially also find it by seeing which side shakes the most. Depending on the mileage, the coils can/do fail from mileage so if you find one coil is bad but the coils are all original with 100-130k miles, I would just do all 4/6 (as well as the plugs) just to nip the failure of the others before it becomes an issue on another cylinder and you end up having the same issue. If one goes out, there's a 100% guarantee the others are next. Just make sure to use the Denso coils - they are expensive compared to the millions of Auto Zone Chinese clones but those things never seem to last. Fine for an old Kia you know is on the way out, but don't do it in a Toyota V6. For plugs, NGK Iridium. The V6 is a PITA to do the back ones because you need to remove a heat shield on the back to do the rear ones, so it's truly a job you get over with while you are there on the V6 engines with a lot of Toyotas.

The other things I see across the board is bad engine mounts which aren't too bad on the ES being FWD, but you only do one at a time and you probably do want to find a way to temporarily lift the engine to make this less difficult. A jack stand with a flat base is sufficient (cut a piece of wood to do this, doesn't need to be straight - I've used scrap wood I've cut with a circular saw as clean as I can get it, or scrap wood/seconds I cut wrong too), especially if you put it in the middle of the oil pan. If the mounts look fine, I would also clean the throttle body, disconnect the battery for 20-30 minutes, and just drive it so it relearns the throttle pattern. If you own a scan tool which can reset and relearn this helps dramatically, but it's not a must have.

There is a known issue on the 2007-2012 ES350: The VVTi rubber lines are known to go bad - if you have one of these, check the line if at all possible. Even if it looks good, do it NOW while you are there if you are able to do it because that job is a VVTi cover off job! You will also need a new gasket too, don't reuse the old one!

If the issue remains, I would look at the possibility of bad gas or a bad fuel pump, even a dirty fuel filter. The fuel pump should be under the rear seat and you usually do not have to drop the tank on Toyota/Lexus (this is not true with a lot of GM trucks, even some passenger cars) and the fuel filter is either part of the fuel pump and serviceable, or in some of the newer ones you change the filter with the fuel pump. Without knowing the year, I can't say for sure what configuration yours is but if the filter is not part of the pump, it's usually under the seats or bottom of the car.