Can protein structure be determined by X-Ray Diffraction in a single image?
Yes. Using a technique called Laue diffraction, it is possible to obtain sufficient data from a single image to solve a protein crystal structure. One example is the time-resolved study of carbonmonoxymyoglobin dissociating by photolysis (DOI: 10.1107/S090904959501661X). This is not the standard single-wavelength technique that is usually used, but uses "white" X-rays with a range of wavelengths available only at synchrotron beams. For example, the BioCARS user facility provides infrastructure for time-resolved crystallography. It is also used in "diffraction before destruction" when working with free electron lasers, see for example Nature Methods 8, page 283 (2011).
The remainder of the answer is about conventional single-wavelength crystallography.
Is the image a composite (where the angle of the point point from the centre is equivalent to the angle of the reading) or is a separate image taken at each angle
The desired structural information (3D electron density in real space) is a Fourier transform of the diffraction data (3D reciprocal space). The word "image" is jargon for a single diffraction image, i.e. the diffraction spots you observe when you point an X-ray beam at a crystal in a certain orientation. Using a different orientation, you get more data (and also measure some spots multiple times).
Are there any other reasons why more images would be required?
The more diffraction images are collected, the higher the completeness and redundancy of the data. Completeness refers to having measured every diffraction spot at least once. Redundancy refers to how often on average a spot was measured, and increasing redundancy increases the quality of the measurement through averaging.