Earlier quoted context omitted.
They'd presumably do this starting from the fringes of the galaxy where the gas is thinner, and with a sufficiently big vehicle (perhaps assembled out of parts that are accelerated separately) that it could withstand some erosion. The sail itself would be discarded after acceleration. Drag on the interstellar medium using magnetic fields could be used for deceleration.
Standing up to millions of years of relativistic erosion seems difficult. Accelerating ISM ions near your destination to slow down would necessarily consume as much energy as it took to get up to speed, and without assist from folks at home (who are anyway a different species by now). You will have plowed through untold tons of relativistic IGM on the way that I suppose you could collect on the way and then fuse to p…
The Local Group is embedded within the WHIM -- Warm/Hot Intergalactic Medium. This is a hot, ionized gas with density maybe a few ions per cubic meter, maybe 100,000x less dense than the ISM in our galaxy. Even better, it's ionized: a magnetic field will enable the ions to be guided around the vehicle without any erosion whatsoever.