I couldn't do it without X is a common way to acknowledge X for its contribution to the success of the speaker's projects, endeavors, career, etc.
This is often heard during acceptance speeches given at awards ceremonies.
I'd like to thank my mom and dad. I couldn't have done it without them!
Neil Patrick Harris (NPH) is a famous actor. Here, it sounds like he is acknowledging the "little people" for their support. Here are two definitions of little people from M-W:
: ordinary people who are not wealthy, famous, or powerful
: people of unusually small size
In this context, I take "little people" to mean people that don't usually get credit for their work, people who work behind the scenes, like writers, costume designers, camera operators, etc. and the many fans.
Further, it might be a play on words. I believe little people is currently a politically correct term for people with dwarfism. Little people are often cast to play elves in Christmas movies. So if there were elves helping NPH in the movie, then he's specifically acknowledging them for their help.
By the way, I couldn't find part of the conversation in this copy of the script. I assume that OP heard correctly though. Throughout their interaction, it seems like NPH is dismissive of Jesus Christ. So "Thanks. Couldn't do it without the little people." does not seem to be directly related to Jesus's statement. He seems to be ignoring Jesus, unwilling to believe that it's him.