Solid tyres for child's tricycle

If you're going for an original-spec rebuild, I'd absolutely stay with white tyres on that bike. However if you just want a kid to ride it, then there's no reason to stay with solid or even white.

3 modern tyres will look fine, and will be more comfortable than a solid tyre. You can sometimes find them in white, or even other colours.

If the rim has no valve hole, then you would have to drill one. Also, depends if the rim has a hook for the tyre bead.... without that a modern tyre won't secure to the rim.