TL/DR: It's not specifically spelled out but the books seem to indicate that their hair/beard will forever appear as it did in life. So, no, he will not need a wig to blend in with dwarven society.
However, see the bottom 2 links to magic items if they wish to alter their appearance
Note: Most of the below comes from 3.5 sources, but I believe the intent stays the same across editions
The developers/writers do not always go into such granular details. So while I have looked in several books I haven't found one that explicitly states what you're looking for. However, we are given some guidance as to their intention:
From (3.5) Monster Manual (pg 250) we get the following: (emphasis mine)
Vampires appear just as they did in life, although their features are often hardened and feral with a predatory look of wolves...they often embrace finery and decadence and may assume the guise of nobility...Despite their human appearance, vampires can be easily recognized, for they cast no shadows and throw no reflections in mirrors.
Then the (3.5) Libris Mortis (pg 14) adds further
Undead that can naturally appear alive also commonly (infiltrate/partake in the society of the living), particularly vampires. In this way, some undead never really leave the society from which they sprang...
So from these 2 excerpts we see that, just like in much of vampire mythology vampires can & do hide themselves amongst the living. We are told that they can be spotted as impostors based off of their lack of reflections & their feral appearance. Although it could be argued their hair may be part of what gives away their disguise it wouldn't seem to fit with blending in to regular society, let alone nobility.
So, where does this leave us?
Arguably, the most likely intended answer is that the vampires hair, much like his body, looks just as it did in life..more specifically on his last day of life when he 1st embraced his gift/curse of undeath. (Hopefully he didn't have a bad haircut at that time)
So what happens if the vampire's hair is damaged? Again, it's not spelled out verbatim but they do tell us what happens to the rest of his body when it's damaged, it heals itself and it heals quickly. To bring in real world facts skin/nails/hair are all made of the same cells so if skin can repair itself it would only make sense that hair does too (without specific directions to the contrary). Additionally, we know that if their hair stayed damaged it would really hinder their ability to blend in as the books say they are ought to do. With the predominantly violent nature of vampires it must be expected that this will happen (multiple times) at some point in their unlife and the fact that the books don't address the ramifications of this would lend itself to the belief that the hair does repair itself.
Lastly, if your vampire wants a different beard/hairstyle their is a magic item Hat of Disguise (3.5 version) or Hat of Disguise 5e version that can easily do this.