Assuming your transcription is correct (because I have not heard the actual audio):
If the key center is D major, then the C chord would make sense as ♭VII. Film scores use this kind of chord all the time, and one name/justification (both unnecessary concepts, by the way, but I can see why you're asking) is modal mixture (or modal interchange). Tends to add some color to an otherwise dull diatonic passage. A variety of other names can arise in other contexts, but to me the chords seem like ♭VII I ii I. If the key is B minor, C major makes sense as the Neapolitan 6th chord (♭II), but resolving it to the III isn't a great example of that chord's resolution.
If the key center is D major
That's unlikely in my opinion (though again, I've not watched/listened to the actual music). It makes waaaay more sense in G major. Think about it: D major has a C♯ in it, and at least in those bars you posted, there are only C♮s. That should be a red flag, but the smoking gun is the melody and chord progression. The melody just sounds like G major, even though it's ambiguous. It wouldn't be a very interesting melody if those same notes were in D major (lots of roots, thirds, fifths of the tonic chord often means less interesting). That, and ♭VII I ii I is a very uncommon progression, whereas its G major counterpart, IV V vi V makes way more sense.
Of course, the best determination of the key center would be to play the original sound that you're analyzing/transcribing, and then listening to which note sounds the most stable, but I'll assume that since you're asking, you probably have some reason not to completely trust your ears on that one (it's a hard skill to do accurately).
And of course, if the key center is in fact G major, the C chord is the IV chord. Not much else to say about that.
Determining the key is actually pretty complicated, and when you get down to it, a "key center" is a musician's construct that allows one to create predictable patterns of tension and release of that tension. Looking beyond all the theory, music is organised sounds, and in your case, it's a melody and chords below it. Who cares whether it's G major or D major? I'm not saying that key centers are useless, or that theory isn't important, but I think that sometimes beginner musicians get too wrapped up in theoretical justifications and labels for every chord they encounter. The better question to ask is usually "What does this chord do?". Does it make you expect another chord? Does it lead you somewhere? Once you've established that, then it makes sense to start looking for labels in order to organise your musical understanding into patterns via terminology and chord-naming.
Bit of a rant there, but I think the important takeaway is to be careful when determining the key. Sometimes, a simpler explanation exists.