Can "ὑπὲρ ἡμῶν ἁμαρτίαν" be translated as "because of our sin"?
No. The Greek language uses noun cases (here nominative, accusative, and genitive) to keep track of each word's role in the sentence. Although it may appear to be viable in English, the proposed translation is disallowed by the case markings of the Greek text. The preposition ὑπὲρ with the meaning "for the sake of" requires a genitive object, which is provided by the pronoun ἡμῶν ("us").1 The accusative ἁμαρτίαν ("sin") therefore has no place in the phrase. It is instead an object2 of the verb ἐποίησεν ("he made").
Can "ἐποίησεν ἵνα" be translated as "He made so that"? (meaning He made it possible that)
This question, in isolation, is whether a ἵνα clause can function as the lone "object" of ποιέω. This is rare (ποιέω generally takes one or two accusatives, as noted above/below), but it happens.3 However, the answer to the first question effectively requires the verb ἐποίησεν govern at least one accusative, so not "he made so that...".
The third problem with the proposed translation is the article τὸν which the OP has left off the beginning of the Greek text:4
τὸν μὴ γνόντα ἁμαρτίαν ὑπὲρ ἡμῶν ἁμαρτίαν ἐποίησεν.....
The article both nominalizes the participle μὴ γνόντα and also removes the morphological ambiguity between nominative and accusative, viz. it must be accusative. The phrase τὸν μὴ γνόντα ἁμαρτίαν ("him who knew no sin") is necessarily an object, and the only candidate verb is ἐποίησεν. The basic syntax of the usual translation is therefore not in question:
τὸν μὴ γνόντα ἁμαρτίαν
him who knew no sin
for our sake
he made [to be] sin
1. Υπερ with the accusative has a different meaning -- "beyond" (see BDAG s.v. ὑπέρ, B.) -- which doesn't make much sense here and certainly doesn't contribute to OP's proposed translation.
2. More precisely, the complement in an object-complement ("double accusative") construction. See Wallace p. 182ff.
3. See BDAG s.v. ποιέω, 2.h.α., providing the example of John 11:37: οὐκ ἐδύνατο οὗτος... ποιῆσαι ἵνα καὶ οὗτος μὴ ἀποθάνῃ; "Can he not ... make [it so] that this man also not have died?"
4. I gather this was inadvertent. If there is a textual variant to that effect which I've missed, please do let me know.