Context. Astrometric monitoring of directly imaged exoplanets allows the study of their orbital parameters and system architectures. Because most directly imaged planets have long orbital periods (>20 AU), accurate astrometry is challenging when based on data acquired on timescales of a few years and usually with different instruments. The LMIRCam camera on the Large Binocular Telescope is being used for the LBT Exozodi Exoplanet Common Hunt (LEECH) survey to search for and characterize young and adolescent exoplanets in L′ band (3.8 μm), including their system architectures. Aims. We first aim to provide a good astrometric calibration of LMIRCam. Then, we derive new astrometry, test the predictions of the orbital model of 8:4:2:1 mean motion resonance proposed for the system, and perform new orbital fitting of the HR 8799 bcde planets. We also present deep limits on a putative fifth planet inside the known planets. Methods. We use observations of HR 8799 and the Θ1 Ori C field obtained during the same run in October 2013. Results. We first characterize the distortion of LMIRCam. We determine a platescale and a true north orientation for the images of 10.707±0.012 mas/pix and -0.430±0.076°, respectively. The errors on the platescale and true north orientation translate into astrometric accuracies at a separation of 1′′ of 1.1 mas and 1.3 mas, respectively. The measurements for all planets agree within 3σ with a predicted ephemeris. The orbital fitting based on the new astrometric measurements favors an architecture for the planetary system based on 8:4:2:1 mean motion resonance. The detection limits allow us to exclude a fifth planet slightly brighter or more massive than HR 8799 b at the location of the 2:1 resonance with HR 8799 e (∼9.5 AU) and about twice as bright as HR 8799 cde at the location of the 3:1 resonance with HR 8799 e (∼7.5 AU).