Branch out a 120VAC From a 240VAC
Yes it can be done, and yes there are conditions on the &quot;how.&quot; Following are a few thoughts that come to mind.

Use the right kind of cordage (ie not building wiring like NM cable). SOOW is a good choice.
The 50 amp circuit is going to require heavy cord (6 gauge). If you don't really need 50 amps (and you probably don't).. you could consider changing out the breaker to 40 amps and using 8 gauge cord instead.
Having multiple sockets on the cart is a little tricky because of the details of joining multiple 6 or 8 gauge conductors. The outlets probably will not be designed for having two conductors landed on each terminal, so daisy-chaining the outlets won't work. You'd have to use connectors like the Polaris brand.
The 120 volt socket on the cart is tricky. It's obviously unsafe to wire that in such a way that it's fed directly from the 50 amp circuit. Unfortunately I don't know the details of how to accomplish this objective while strictly meeting code. I confess that I've put such a socket on my welder cart, too.. I protected it with a 20 amp fuse installed inside the junction box. A panel-mounted fuse holder or circuit breaker would have been a nicer solution.