How could a generation ship "steal" energy or resources from star systems which it passed without changing velocity?
It could deploy a swarm of von Neumann drones well ahead of the ship to collect resources and then blast them up to speed so that they are ready for collection when the generation ship cruises by.
For example, if the ship is going 20% c (the speed of light) and it sends out a probes ahead that are going 22% c, then for a system that's 10 light years away that gives the probes years to arrive, replicate, and then start sending out resources at the required velocity.
It could deploy wide solar panels to adsorb star light and gain energy.
It could use a Bussard collector to gather interplanetary matter, which should be many times denser than interstellar matter, and be careful to maintain the same speed and direction, making corrections and adjustments as necessary if it slows down while collecting the interplanetary matter.
It could deploy faster smaller, faster, ships ahead of time to do the job. The harvester ships are launched from an already fast ship so they get a head start on their acceleration to travel faster than the generation ship. They need only be able to decelerate in time to arrive at the target.
But picking up the resources is trickier. If the harvester ships are able to accelerate under their own power faster than the generation ship they could catch up to the generation ship long after it has already passed the target.
If not, things get...complicated. It starts to become a matter of degree as to how slow can something be traveling and still be safely caught by the generation ship.
Obviously, this would be a generation spanning endeavor for each target.
If you are stealing matter, in the ideal case you start with an inelastic collision. Then, you must accelerate the new combined mass (depending on what % of your ship's mass they are picking up, and how tight your "without changing velocity" constraint is). It isn't a super tricky calculation -- since this is a generation ship, you'll have a class of highschool students, hopefully the brighter ones will be able to solve it with little difficulty if you pose it for them properly.
If you are "stealing" energy, probably just use solar panels (I'd argue that stealing energy that was just going to be blasted into space isn't really stealing). Technically the photons will impart a tiny change in velocity, if you want to be really strict about the requirement to not change velocity, you'll have to accelerate to compensate. But note that these photons are going to hit you anyway.
The video game Elite (1984) had the concept of Fuel scoops which would scoop up ambient hydrogen when flying through a star system. something like that.
Resources heavier than H2 would be more difficult, as you'd have to overcome inertia, but again, a scooping concept would probably work best, trace amount of resources through dust clouds, or a system that could crush small space debris into dust, and then stored would be practical