How many times does the moon orbit the sun in one year?
It is little less than once. Presume a new year day is a new moon day. Then next new year day will not be new moon day. The position of Moon would be a bit lagging or leading compared to the position a year ago with respect to Earth. In this way it shall be more than 365 days. But this matter is very complicated to be considered. So for a casual answer it is same as that of Earth.
the moon only orbits the sun one time for every time that the earth rotates, however if you want to get technical, then because the moon orbits the earth it would take the moon a fraction of a second more or less times to orbit the sun however if you want to get that technical then you have to realize that you are currently traveling around the center of the earth at nearly 24,000 miles per our, flying around the sun at about 18.5 miles per second thus the difference would be measured in nano seconds