It should indeed be noted that Vega is not a constellation, but an individual star; currently, there is no constellation named Vega, so there cannot be an individual star named Delta Vega, either. (Or a planet named after its mother star, as the situation with these Greek-letter Trek planets probably usually is.)
But many of the stars in our galaxy are hidden from our view by dust clouds and the like. Warp drive would change that overnight, as even a very short "side step" would allow us to peek past the obstacles and see new stars. Possibly, then, a cluster of stars in the exact direction of Vega would be added into a constellation named Vega, with Alpha Vega just 25 ly from Earth but Delta Vega more like 1,800 ly away.
On the other hand, the reboot movie offers an interesting alternative. There, a planet named Delta Vega is located in a completely different part of the galaxy. No, we don't know exactly where, but it's close to Vulcan, which lies neither in the direction of the star Vega nor close to the "edge" of the galaxy, by any definition of edge.
If Delta Vega was named after Nero messed up the timeline, there's one way two places in different directions would have equal chances for the name. Rather than the fourth-brightest star in the constellation Vega, the planet could be the fourth owned by the VEGA corporation, just like Dytallix B sounds like it would be the second one added to the Dytallix business empire... Perhaps VEGA was bolder in a timeline where giant Romulan starships didn't materialize out of nowhere and blow up Federation hardware, and established Delta VEGA farther out?