Right, so I've just entered the forgotten vale and it looks to be quite a huge place so naturally I figured I would use Arvak.....
But no, it will not allow me to summon him? Ok, I guess I'm technically in a cave so as a vampire I should be able to
regen life, stamina and magica despite it being day time and yet I don't.

So is the forgotten vale an outdoor area meaning I should be able to summon Arvak and not regen points?
Or is it an indoor area?

It seems like its both, and not for any good reason.

You can't summon Arvak in towns either. Just because its outside doesn't mean you can summon a horse. If the Forgotten Vale was treated exactly as an outdoor area then if you fast traveled within it your normal horse either from the stables or possible shadowmere could suddenly appear.

The Vale can be fast traveled to though, right?

You can fast travel there....

So where does your horse go if you fast travel there with it?

The same as if you travel to a town. I beleive the Forgotton Vale is treated a lot like Blackreach.

If that is the case then I should be able to summon Arvak as I'm able too in Blackreach

Arvak cannot be summoned in the Vale because the Vale is actually a worldspace the spell is restricted from working in.

Which is...?

If you've been to the Forgotten Vale you've been to Blackreach. Remember that HUGE underground Cavern you visted through a Dwemer Ruin while looking for the dragonscroll.

Sorry, I meant where does your horse go if you fast travel to the Vale?

Where you would have been forced to leave your horse behind when you first visted.

Gameplay wise the Vale is outside the Tamrielic world space the Arvak spell seems to be restricted to the Tamriel World space.

Isn't the Soul cain in oblivion therefore not in Tamriel?

The Vale is much similar to Blackreach in how it works. The only reason you're able to Summon Arvak in Blackreach is because of a simple oversight, I'm positive it was not intended. The Vale was added in the same DLC as Arvak was added (obviously) so it makes sense how the devs could remember to prevent it from working in the Vale but forget about Blackreach entirely.

Well the fact that you find him there makes that an exception..I suppose that you shouldn't be able to summon him in Blacreach as well but you can due to either an oversight or bug.

I believe Convenient Horses allows you to summon any horse inside the Vale, including Arvak. All our horses are now outside Fort Dawnguard. I don't actually use Arvak myself, Serana does, and when we all mount our horses outside the fort I seem to remember that Serana summons Arvak too.