Note: this is what I’ve been taught in high-school. Generally, grammar in high-school isn’t too accurate, but I hope it'll do here.
Tl;dr: you need a "que" to mark a change in the subject. It's optional in the cases where the subject of the subordinate is present (both as a subject or as an object) in the main clause, being necessary if the subordinate clause is finite, and unnecessary if it's nonfinite.
First of all, pretty much all finite (personal) subordinate clauses require an introductory “que” in Spanish. Sometimes that “que” works as a pronoun, and others it is just a linker with no semantic content. The sentence at hand would be an example of the latter.
Secondly, it is note-worthy that if you translated your sentence to Spanish word by word, you’d get something like:
Desarrollaría un sistema que haría a los niños querer aprender.
Note how the verb “want” is an infinitive in English, not an inflected form. This is made evident by changing the subject to a third person singular:
I would develop a system that would make her want [not wants] to learn.
The problem with the literal translation is that whereas it is quite common to add a subject to impersonal forms in English, it is not so in Spanish. The only case where a verb in its impersonal form would take a subject in Spanish is when the subject in the main clause is the same as the one in the subordinate clause, so it becomes an implicit subject, or when there’s a noun or a pronoun in the main clause that marks the subject of the subordinate clause. Compare:
Quiero cantar. (I want to sing)
Quiero que cantes. (I want you to sing)
In the first case, the subject of the main clause ("quiero") is “yo”, so the subordinate verb (“cantar”) is in an impersonal form (an infinitive) with the same implicit subject (I want [me] to sing). In the second case, the subject of the subordinate clause is “tú”, which is not the same as the one in the main clause (“yo”). Since impersonal forms can only take subjects that are present in the main clause, and given that “tú” isn’t present in “quiero”, you must use a personal form (“cantes”), even though you would use an impersonal form in English (“to sing”). Now let’s take a look at a different and grammatically relevant example:
Me gustaría cantar. (I’d like to sing)
Me gustaría que cantes. (I’d like you to sing)
The subject of the main clause is the same in both sentences: the subordinate clause (keep in mind in Spanish, the subject of the verb “like” is the object liked, not the one who likes). In the first sentence the subject of the impersonal form of the verb “cantar” is an implicit “yo”. Here, the subject of the main clause isn’t “yo”; however, the personal pronoun “me” works as a marker of the implicit subject of the subordinate clause. That’s why you can use an impersonal form. In the second sentence, there’s no pronoun whatsoever marking the subject of the subordinate clause (“tú”, “te”), so an impersonal form wouldn’t work here.
Now you now when you need to use the impersonal form of the subordinated verb and when not to. Once you know you need to conjugate it, you also know you need a “que” working as a linker (see the first paragraph).
Now, the sentence you want to translate is a specially complicated one. Grammarians disagree about the function of subordinate clauses introduced by the verbs “hacer” and “dejar”. I actually asked a related question not too long ago, and from the answers I got it seems some grammarians consider it to be a complemento predicativo. I don’t fully agree with that, but what I can say for sure is that it is not a direct object. However, the aforementioned rule applies: if there’s any pronoun/noun in the main clause marking the subject of the subordinate one, or both clauses have the same subject, then you can use an impersonal verb form and therefore you don’t need a “que”. Otherwise, you need a personal form and therefore you need a “que”.
Your sentence, if translated literally (i.e., keeping the verb impersonal), isn’t grammatically wrong because you have mentioned the implicit subject of the subordinate clause (los niños) in the main clause ("a los niños", which can be substituted by “les”). However, it doesn’t sound just as natural as the conjugated version (which requires a “que”, as explained above):
Desarrollaría un sistema que haría que los niños quieran aprender.