Well, I have to say this is a novel idea. Just that - something that would fit in a novel! Some post-apocalyptic adventure: "a girl and her garden." :)
But for fun and the sake of experimenting, let's give it a whorl.
There are plenty of veggies which are capable of surviving all sorts of adversities, but your biggest problem, I think, would not be the vibration or the heat alone, but rather, the rapid fluctuation in temperatures that driving long distance could subject you to. Doubtful most veggies would set fruit under the accumulation of those conditions. They might bloom, but the blossoms would most likely fall off right afterwards. So, goodbye to the bulk of crops: tomatoes, peppers, eggplants, squash, etc...Roots wouldn't be dependent on blossom set, so you have radishes, potatoes, carrots, turnips, etc. These would all need a good depth of soil, of course, and I think they probably WOULD be susceptible to the vibration - meaning the roots might not thicken into anything useable. OK - so leaf crops: lettuce is very sensitive to change in temps, and will bolt if things aren't good for it - meaning it will stop producing edible leaves and try to set seed, something plants in stress will often do.
All in all, not looking good. That's not to say you shouldn't try it. Just don't put much money into it! You'd be better off using the trailer to haul extra food and water. But hey! If you try it and it works, you could write a book about it. :) Good luck.

I too have thought about this concept and would love to hear more about your trials and tribulations...

Some ideas to counter the above points would be to try to build an anti-vibration bed of sorts - a high-tech solution may be in order with active damping.

Another approach may be to use a soiless medium and use some sort of hydroponic setup within an enclosed trailer; perhaps one with a clear roof or light pipes.

I have thought that if you live on a house-boat, a portable (slow-speed) floating garden would work, perhaps using techniques like chinampa gardens like those used in Cuba.

Best of luck!

Hi
A very interesting idea!!lol Would suspect that the cost would be prohibitive ?? Using the trailer for storing food
that is already pakaged would probably be far more manageable?? They have a garden on the space station
Only advantage is 24 hour sunlight and I'd bet those carrots and radishes cost less than 250 dollars each lol good luck with whatever you decide. I also would like to hear the results?? gary