Reaching all regions of Earth, low Earth orbit (LEO) satellites can harvest delay-tolerant data from remotely located users on Earth without ground infrastructure. This work aims to assess a data harvest network architecture where users generate data and LEO satellites harvest data from users when passing by. By developing a novel stochastic geometry Cox point process model that simultaneously generates orbits and the motion of LEO satellite harvesters on them, we analyze key performance indices of such a network by deriving the following: (i) the average fraction of time that the typical user is served by LEO satellite harvesters, (ii) the average amount of data uploaded per each satellite pass, (iii) the maximum harvesting capacity of the proposed network model, and (iv) the delay distribution in the proposed network. These key metrics are given as functions of key network variables such as $λ$ the mean number of orbits and $μ$ the mean number of satellites per orbit. Providing rich comprehensive analytical results and practical interpretations of these results, this work assesses the potential of the delay-tolerant use of LEO satellites and also serves as a versatile framework to analyze, design, and optimize delay-tolerant LEO satellite networks.