#Almost no one under these conditions will travel
(average speed is 4 times the speed of light)
That means a year of travel from here to Earth's nearest not-the-sun star. Calculating time ship-board is meaningless, as once you've got FTL travel time dilation calculations don't work. Usually in these situations Ship-Time and Universe-Time flow 1:1 (because fiction!), which would mean a year on board for those people making the trip. Not exactly...thrilling stuff. Just keeping the crew fed is going to require a significant amount of cargo space.
At such distances and time scales, the only communication (e.g. letters) are going to be of sporadic personal or scientific in nature ("colony's going well, how have you been?"), albeit expensive (most letters will likely travel as negligible bulk on a cargo ship carrying a few million tons of something else). You won't have an effective "galactic" government that can control and enforce laws: remember the turnaround time on stuff like "help we're being invaded by aliens" is 2 years (even if the crew experiences no time!) and things like "You aren't following our laws, prepare to be invaded" are virtually meaningless. What's central authority going to do, show up in 4 years to put down a rebellion that's been over for three?
You could have trade, but it'd be for either resources you literally can't get (but if a delivery goes missing you'll survive: next one's not coming for a year). Even at 4:1 time dilation it would still be 3 months ship-board journey for every 4 light years traveled, which is almost comparable to current ocean shipping (where are small, often can't even communicate in a single language, and live in comparative squalor; its been compared to living in prison).
but have only densely colonised the star systems with 15-20 lightyears of Sol.
I...hate to tell you this, but that's like 4 potentially habitable star systems in that range. There's about 150 stars within that distance from Earth but most are red or brown dwarfs:
Nearby Celestial Objects by Type, Number, and Mass
Spectral or Number Sum of min
Luminosity Type within 20ly solar masses
O - Blue Stars 0 0.00
B - Blue White Stars 0 0.00
A - Bluish White Stars 2 3.84
F - Yellowish White Stars 1 1.50
G - Yellow-Orange Stars 7 6.76
K - Orange-Red Stars 15 11.84
M - Red Stars 88-90 21.32
M,L,T, Y - Brown Dwarfs 15-30 <1.20
D - White Dwarfs 7 4.30
Total Objects > 150 > 51.46
The 8 yellow stars are pretty much your only targets. Using Wikipedia's list of exoplanets we find 5 "potentially habitable" ones within 15 light years and 2 more out to 20 and another 3 ranging out to 30ly. That's pretty sparse with grueling ~8-12 year travel times (one side of inhabited space to the other); even between nearest-neighbors it might be as much as 4 years.
The only fiction I can think of that featured space travel on these kinds of time scales is The Deepness in the Sky where the Qeng Ho (humans of primarily Chinese decent) slow-boat it from system to system, trading where they can, helping inhabited planets rebuild after the collapse of civilization where they can't (because in a few hundred years they will be worth trading with again). And they primarily trade in technology. Not gadgets, but blueprints. Knowledge. They'll buy and transport things too, but those are purchased by the Qeng Ho for use by the Qeng Ho.
They do it as a safety net against humanity nuking itself into oblivion and going extinct, not because its profitable. As a result, they have a massive knowledgebase capable of kick-starting any planet to reasonable levels of technological comfort in decades (instead of millennia). Doesn't matter what you have access to, the Qeng Ho can work out an action plan, give you the right tech, and come back in a hundred years to see what Neat Stuff you invented while they were gone.
