Sorry if this this doesn't really fit this forum, but WorldBuilding does not like questions like these. I cannot find the minimum density of hydrogen a Bussard ramjet (an interstellar ship propelled by hydrogen collected from the interstellar medium) needs to work, or the densities of these certain areas(galactic core, thicker arms, Large Magellanic Cloud, Small Magellanic Cloud, Magellanic Bridge, and the Magellanic stream, and intergalactic space). I am interested if this is a viable mode of transportation in different parts of the universe relatively near to us.
Yes it's theoretical, but on some levels, estimates can be calculated. (probably better for space-ex than here), but it's a fun design to think about.
The basic design, the ship generates a magnetic field and catches atoms in space as it moves. I've read that the "catch" might be about a square mile in area, so you can calculate how much hydrogen and alpha particles and other particles (far less) and electrons (some). As the ship picks up speed it's catch rate would increase.
There's two primary problems. The materials that it would catch in space, mostly charged particles. Protons and Alpha Particles, they are not easy to fuse. There are much better fusion materials available if you make your own deuterium for example, or lithium deuteride for easier storage, or 3He. Fusing mostly protons and alpha particles is very hard and might never be practical. You could in theory calculate the energy output - 4 protons into 1 helium yeilds 0.7% mass to energy ratio, which isn't bad, but won't get you close to the speed of light.
The other problem is that the magnetic catch that the ship generates, you have to take into account every particle it encounters and factor in the drag. If each proton can deliver 0.7% of it's mass into energy, there becomes a velocity where that proton hits the ship where you lose about that much upon impact.
A 3rd impracticality is that space is very empty and hydrogen is very light and quite abundant. It doesn't make much sense to collect it in empty space when you can simply carry your own.
Bottom line is that it's probably not a useful design. Could it work? Sure. It would work best riding a stream of particles behind you, like riding a jet from a distant solar flare or supernova. You'd still need to solve the the hydrogen fusion problem and if you solve that, you could probably design better ways to fly than a ramjet.
So, yes, it could work. Solar panels can drive a car and sails can move a battleship, but neither is particularly fast.