You could check out our theory & hypothesis paper where we studied 3 infants longitudinally using eye tracking (and compared to 9-month-olds). We started when they were around 2 months old. We have had difficulty calibrating the device to their eyes and obtaining data before 2 months of age. Here's the citation:

Corbetta, D., Thurman, S. L., Wiener, R., Guan, Y., & Williams, J. L. (2014). Mapping the feel of the arm with the sight of the object: On the embodied origins of infant reaching. Frontiers in Psychology, 5(576), 1-18. doi: 10.3389/fpsyg.2014.00576

We have measured eye and head movements in a series of experiments.The youngest was 10 days old. In an optimal setup there are no problems to measure infants 6 we of age. publications: see pubmed, vonh often C and rosander K

I have done various eye-tracking work with 4-month olds with no real difficulty and I know that plenty of other researchers have used younger subjects. The model of the eye-tracker may make the most difference. I have used both ASL pan-tilt models and Tobii models to eye-track at 4 months. The only problem I've faced at that age is that the chubby cheeks of the babies sometimes cuts off the bottom of the pupil making it difficult to pick up the corneal reflection for proper discrimination (something you can easily see on the ASL). When that happens, waiting until the baby stops smiling or having the infant sit more upright helps. 

We have successfully tested many infants with a remote eye tracker, the youngest was a couple of weeks old. When you attach the eye tracker's monitor to a flexible (swing) arm, gaze signal reception can be improved, even in infants lying in prams. The problem of calibration can be resolved by performing a post-calibration, which entails aligning the gaze positions with the predefined target positions, after the data collection.

The method we use is described in Pel et al., 2010; J Neuroscience Methods.

There are several published studies that reported on eye tracking in infants < 6 months, e.g.:

Ahtola et al 2014 PLOS One

Jones et al 2014 IOVS

review by Aslin & McMurray 2004