If the Sun disappeared, could some planets form a new orbital system? If the sun were to suddenly disappear, the planets would continue to travel tangentially to their former orbits. (This I know from this answer to a somewhat related question here.)
In such a scenario, is it possible (however unlikely) that a new orbital system would form from two or more of the now-rogue planets?  For example, is it physically and mathematically possible that Earth might end up getting captured by Jupiter and thereby become one of its moons?  Or do the size and speed of the planets completely rule out such a scenario, even if the sun disappeared at such a time that the planets' new trajectories would cross?
The issue here is whether pairs of planets can become gravitationally bound to each other.
In the two-body problem the trajectories or orbits are ellipses (bound orbits), parabolas and hyperbolas (unbound). For all practical purposes, an encounter looks like they start out at infinity with some finite speed, approach each other, and then maybe fly away or remain bound. This situation (finite speed when far away) corresponds to a hyperbola, so the generic case will not form a new bound system: some of that energy needs to be shed somehow.
One way this can happen is an inelastic collision, i.e. a merger. A less extreme case is a close encounter where tidal forces dissipate some of the kinetic energy. But the kinetic energy of planets is large, so a single encounter is unlikely to dissipate enough to make them bound.
Another important possibility is a three-body encounter. This is how comets get into short-periodic orbits when falling into the solar system: an encounter with Jupiter (or some other planet) leads to a gravitational &quot;assist&quot; that makes it bound with the sun.
In your scenario where the rogue planets fly off along their tangents it is very unlikely that any start out bound to each other since planetary orbital velocities around the sun are much bigger than the orbital velocities of a bound planet-planet pair at a large distance. Hence they will just fly apart. In theory they might have an encounter before leaving the solar system vicinity (e.g. Mercury by sheer chance sweeping past Jupiter) and that leaves a tiny chance of a three-body interaction with moons or a merger, but the chances are truly tiny. The generic case is just scattering.
Addendum: Some estimates
For a body to be bound to another body of mass $M$ it need to move slower relative to it than escape velocity $v_{esc}=\sqrt{2GM/r}$. If we assume planets move in circles we can then take their orbital velocities for all possible positions along the orbit, calculate the relative velocity, and compare it to the escape velocity.
The result is this diagram, where I have plotted the separation on the x-axis and how many times escape velocity the relative velocity is on the y-axis. The numbers denote which pair of planets I am comparing. The blue horizontal line is $\Delta v=v_{esc}$.


Looking just at the lowest relative velocities shows that the closest to being potentially bound is Saturn to Jupiter were the sun vanish just as they are moving in parallel, but even in this case Saturn has several times Jupiter escape velocity.
One could complicate it with inclinations and elliptic orbits, but it will not change the qualitative picture: the planets are not even close to being bound to each other in the absence of the sun.