It depends on how much weight you have in the tool box. If you keep a lot of stuff in there, it's not gonna wiggle with a mini-lathe on it. Tool boxes ALL wiggle in the store. They firm up with weight in them. (unless it's too cheap and the drawer slides don't work any more... Don't go filling them with concrete.) But honestly, if it's loaded, the casters aren't going to wiggle under the crushing forces applied by your hand on the compound handle on a mini lathe. With any of them, you're going to want a heavy top. Or maybe a heavy base plate for the lathe. Probably needn't be the whole box top.
Do you have it yet? I've eyeballed that tool box, and I'd put it on par with the Harbor Freight Yukon line. If you could swing the extra hundred and twenty bucks, I'd suggest looking at the US General 42X22. All of these budget tool boxes, no matter where you get them, they'll hold a lathe. I am talking about the rest of it. That US general, that's a step up from that Home Depot box, and a step up from the Yukon box, but the drawer configuration is undoubtedly different, and it is more bucks, but once you load it up and have some weight on any of them, that's wen they don't all feel the same anymore. IMO it's 50 percent better box, at a 25 percent higher price, and a good return on those extra bucks. Or if you can't swing that, honestly any of them would hold the lathe. It's the drawers and slides that get better.
Anyhow.... If you've got some weight to put in the box, the mini-lathe isn't going to throw it around. Set it up, put it where it goes, and use it. I'll bet it's fine without doing much to it at all.
