The better question would be, how much would it cost? If you're starting with or using standard height cabinets, it would be somewhat difficult to do this cabinet. In order to make it work & allow the doors to match up with adjoining cabinets, you would need to make it shorter from the top, not the bottom. Most cabinetry lines I know, have some sort of a rail crossing the width of the cabinet. This would be true with frame & frameless cabinetry. Unless you go with a cabinet line that allows customizing, you'd probably need to go to a custom shop.

Then there is the factor of how true your floors are to accommodate a 4-wheel cart. You may have 3 wheels touching and the 4th floating in air allowing the cabinet to rock when used. There's also how true your floor is under the counter space you would keep it.

Yes, it can be made. I've made them myself when I had my custom cabinetry business. It's a nice idea. just not readily available for a standard kitchen. I thrived on items & kitchens that no one could find elsewhere.

I have on either side of my range moveable cabinets with butcher block tops I have a catering biz and often have 5 or 6 people in my kitchen these allow the cabinets to actually be used in another space for prep or pastry prep. But they are the same height as my other cabinets I am not sure how useful a lower cabinet would be . IMO if you are getting custom cabinets this should not be all that difficult to have done.

Not hard at all for your local cabinetmaker..

Another thing to consider... when the cart cabinet is relocated, you'll be exposing the sides of adjacent cabinets, and the back wall. I'd want these to be neatly finished, whereas normally they may be unpainted, with exposed screws, etc. This could be an additional expense, or you may not care.

Why do you want movable cabinets?

Obviously it can be accomplished - almost anything is possible if you have the money third the right person.

But can you get what you need accomplished more simply.

I once had a portable dish washer that slid under the kitchen counter like you pictured. Had no problem with it and can see it being a cupboard like you want. Would be very handy for me. As for seeing the sides of the cabinets and the back wall when it is pulled out...no problem. Just finish the cabinet sides like the rest and the back wall can be painted like the walls. I wouldn't think it would be expensive because looks like you could use an ordinary cabinet where the rollers would take the place of the toe space. Fitting under your cabinet might be a problem but I can see smaller rollers or inset rollers Go for it!

A popular option with very small kitchens with limited counter space. Unlike a rolling cart pushed into a corner taking up valuable space, this option has the storage with extra countertop when needed.

I used to have a cabin with under cabinets on wheels in the garage. Move the truck, Pull out the cabinets, sit on a stool at the counter.

I could see using the rolling cabinet as an island. I used the rolling cabinets as additional workspace.

I have seen commercially available garage? undercounter cabinets on wheels in hardware/home improvement stores.

having cabinets on wheels is not difficult.

I think it a fabulous idea! Cant be that expensive.

No, cabinet with a reinforced bottom and casters isn't that expensive. A cheap hack of that can be done. It just looks like a cheap hack. It's the building it in to look seamless that's expensive. It's finishing it on all 4 sides, plus everything in it's alcove being finished that is expensive. It's getting all of the clearances exactlyneight for everything that takes talent and money.

It's not rocket surgery.

Use finished plywood for the carcass and the sides are all prefinished.

An example:

Other possibilities for more work room are a fold down, or pull out countertop.
https://food-hacks.wonderhowto.com/how-to/countertop-hacking-5-ways-increase-your-workspace-tiny-kitchen-0151098/