Here is a theoretical physicist's answer to this problem (see below for what I mean by this). This should be adequate to get a handle on the relative positions of satellites when they are distant from each other, and when the time after some known position & velocity of the satellites is not too long.
The theoretical physicist's approach
First of all make some assumptions:
- the Earth is a spherical mass distribution;
- the Earth is very massive compared to the satellites;
- we can ignore the Moon and other gravitational perturbations;
- the satellites are orbiting in vacuum;
- in general we can ignore any other perturbations, gravitational or otherwise.
The second of these is probably true enough: none of the others will be the case if you want really good answers.
So, OK, then the orbit of any satellite simply involves solving the Newtonian two-body problem for gravity, which was done long ago, and for which there is a closed-form solution. Since the Earth is assumed to be very massive compared to the satellites the barycentre of the system can be treated as the centre of the Earth.
If we know the position and velocity of a satellite relative to the Earth at any point in time then we can now use the closed-form solution to tell us its orbit for all time. And we can do that for any number of satellites.
The problem of knowing the relative positions now becomes computing the vector of one satellite from another. Chances are the natural way to solve the orbit problem involved using spherical-polar coordinates based on the centre of the Earth: to make the computation of the vector of one satellite from another simple the obvious thing to do is to change to Cartesian coordinates based on the centre of the Earth (or, better, on one of the satellites), whereupon the relative vectors just drop out. The coordinate transformation from spherical polars to Cartesian has a closed-form of course, so all of this has a closed-form, & we're done.
The proper approach
So, OK, I said that was the theoretical physicist's answer. In particular I made a bunch of 'spherical cow' assumptions, none of which are true. In real life the mass-distribution of the Earth is not quite spherical (or even constant with time), the Moon & other bodies do perturb the orbits, satellites are not orbiting in complete vacuum and so on and so on. That means that the computation of actual orbits – orbits good enough for things like collision prediction &c – will involve both perturbations from the ideal orbits discussed above, and probably just hairy numerical approaches with data being fed in from time to time from observed positions of the objects. None of this will have closed-form answers in general.
However, if you assume that you have some, probably numerical tool which tells you where satellites are, then based on that it's fairly easy to compute relative positions. What you know is both where the satellites are and the coordinate system where that information is expressed, so what you need to be able to do is change coordinates to a suitable Cartesian coordinate system based on one of the satellites, in which the position vectors simply are the coordinates of of the other satellite.