Indirect, reflexive
 

This is a sentence from the Easy Spanish Step-by-step "La madre quiere que los niños pongan la mesa." SHOULD THIS BE WRITTEN AS "La madre (les) quiere a los ninos que (se) pongan la mesa"?

It's just a different way of speaking but has the same meaning? Would I be understood if I speak to a Spanish person using the latter sentence? Also, is "le/les" sometimes can be omitted?

No, it was written correctly in the lesson.

poner la mesa = set the table
The verb poner is not pronominal, and the subject is known (los niños), so there's no reason to use the pronoun se.

Mother wants ... (what mother wants goes here).
If what the mother wants is for someone else to do something, a noun clause, introduced with the conjunction que, must follow the verb, and there will always be a change in subject (from mother to children, in this case) in the noun clause. (The verb in the subordinate clause will always be in the subjunctive mood.)