Electrical – Would a 20 amp 240 volt panel from the main panel be enough for a laundry room

If you're in the United States, a grounding conductor is defiantly required for all new work.

If the system was installed before grounding was required, you're not required to update the system and add grounding when the code changes. However, if you add anything to the system after the code change, you must do so in a code compliant manner. This might mean bringing the entire system up to current codes, or at least enough of it to allow the new work to be code compliant.

You might be able to simply install a proper grounding electrode at the building, and use that to ground the circuits on the secondary side of the transformer. You'll have to contact your local government to find out if that's adequate, and also to have the work inspected to insure it's done properly.

While the panel is rated 125 amp, you have a 100 amp service. You also have an electric dryer and water heater in addition to the usual lights and outlets. Adding a 66 amp load to all of this is definitely overloading the service.

If you went ahead and installed this welder, you won't burn down the house (at least not from electrical loads, from welding is another story) but you would get nuisance tripping. Every time you struck an arc, the lights would dim. The resulting voltage drop could damage electronic equipment such as computers and TVs, as well as anything else with electronics, which is virtually everything these days.