Whom vs who

No, it is not correct. You need "whom" in that sentence. Whom is objective, who is subjective. The subject of both clauses is "we." The first object is whom. It becomes easier to find this by making a simple sentence and then substituting the third person singular, masculine. "We choose him." That's objective, therefore you need whom.

Thanks, Setanta. Much appreciated.

There’s a continuing debate in English usage about when you should use who and when to use whom. According to the rules of formal grammar, who should be used in the subject position in a sentence, while whom should be used in the object position, and also after a preposition. For example:

Who made this decision? [here, who is the subject of the sentence]

Whom do you think we should support? [here, whom is the object of support]

To whom do you wish to speak? [here, whom is following the preposition to]

Some people do still follow these rules but there are many more who never use whom at all. The normal practice in current English is to use who in all contexts, i.e.:

Who do you think we should support?
Who do you wish to speak to?

(Oxford dictionaries)

I agree with "whom" as a general ruling in this case.

However, it should also be said that a lot of twaddle is often spouted about "English Usage" because appropriate usage depends on context. What is acceptable in local spoken dynamic usage (in which even the concept of "a sentence" is nebulous) can differ markedly from what is acceptable in general written static usage, including the formal rules offered as a pedagogical "fix" for non-native speakers.
Much of the so-called "debate" about usage in this forum, including appeals to authority, is generated by ignorance of such contextual considerations.

The sentence is indeed correct. Setanta, sadly, yet again, for the umpteenth time is not.