I am using a program to learn Esperanto and I have to translate the sentence: "The boys first asked for three cakes." I managed to figure out, "La knaboj unue petis -- trijn kukojn."

What is the missing word I'm looking for? Does anyone mind explaining the differences between "por / pro / cxar / dum?"

I personally would use "La knaboj petis tri kukojn."

Or if "peti" is too strong a word, maybe "La knaboj demandis ĉu ili povus havi tri kukojn."

For your second question:
"por" equals more or less the English preposition "for".
"pro" equals more of less the English "due to", "because of". It gives the reason.
"cxar" gives the reason, but introduces a subclause, like English "because".
"dum" expresses contemporarity, i.e. two actions take place at the same time, like English "while" (introducing a subclause) or "during" (introducing a nominal phrase).

So does it work if I write, "La knaboj petis unue por trijn kukojn?"

If the program marked you wrong, the problem was "trijn". It has to be "tri". Numbers don't vary in their base form, so *trij and *trin are forms you will never see.

If you add the adjective -a or the adverb -e to numbers, only then can they act like normal adverbs. So you can say "trie" (thirdly) or "tria" (third), plus normal adjective forms of "third" (tria/trian/triaj/triajn). Those forms triaj/triajn will be relatively rare, only because it is relatively frequently that you say "I'll have the third cake" or "I won the third place" but "I'll have the third cakes" or "I won the third places" are not so common.

It's true that "peti" can mean "ask for". But tell yourself this - "peti" can also mean "request", so if you worded this in English as "The boys requested three cakes", there is no "for" whatsoever! Try to remember that when you translate sentences; not every word might get translated, and in many cases, you might even be able to re-word a sentence in English to eliminate a certain word, especially little words like for, to, by, etc. Try to translate meanings rather than individual words, and if Esperanto's meaning doesn't match up word for word with the exact English words in the sample sentence, that's ok.

Foreigner:So does it work if I write, "La knaboj petis unue por trijn kukojn?"

No, the correct sentece is "La knaboj unue petis tri kukojn" or "La knaboj petis unue tri kukojn" (the first variant is probably more common).

You know how it sounds a little weird, like you have an extra word, if you say "The boys requested for three cakes", whereas you'd expect "The boys requested three cakes"? That's how it sounds if you say "petis por". (and if you did have por, you'd lose the -n ending, because you don't put -n after most prepositions - so in that case it would have to be "La knaboj unue petis por tri kukoj", but that would be a strange and rare way to say it, though it is not a good translation of the English).