Just got it. As regan pointed out, I had to add the user to the sudoers group. But the main reason was I’d forgotten to update the repositories cache, so apt-get couldn’t find the sudo package. It’s working now. Here’s the completed code:
FROM ubuntu:12.04
RUN apt-get update && \
apt-get -y install sudo
RUN useradd -m docker && echo "docker:docker" | chpasswd && adduser docker sudo
USER docker
CMD /bin/bash