According to the Enterprise episodes "Home" and "Daedalus" Vulcan's star is about 16.0 to 17.0 light years from Earth.
In "Home" Tucker and T"Pol are on Vulcan when:
T'POL: Trip, I have to do this for many reasons.
TUCKER: And how am I supposed to take this?
T'POL: I'm sorry.
TUCKER: You're sorry. You brought me sixteen light years just to watch you get married to someone you barely know.
EMORY: Sub-quantum teleportation. You step on to a transporter on Earth, a few seconds later, you're on Vulcan.
TUCKER: That's over sixteen light years.
Therefore the distance from Earth to Vulcan's star system must be between 16.0 and 17.0 light years.
epsilon Eridani is about 10.48 light years from Earth and not in that range.
The only star systems in that range that don't contain only red dwarfs, white dwarfs, brown dwarfs, and other unsuitable stars are: 40 Eridani or Omicron 2 Eridanai, 70 Ophiuchi, and Altair.
In "Operation Annihilate" Spock said:
SPOCK: An hereditary trait, Captain. The brightness of the Vulcan sun has caused the development of an inner eyelid, which acts as a shield against high-intensity light. Totally instinctive, Doctor. We tend to ignore it, as you ignore your own appendix.
Since Spock mentioned the brightness of "the Vulcan sun", we may assume that only one stars is close enough to have a visible disc and contribute much heat and light to Vulcan.
Altair is the only single star of the three best choices on the list, and it has the most intense light of any main sequence star on the list.
But in "Amok time" the Enterprise was being sent to Altair VI at the same time Spook needed to go to Vulcan. If Vulcan was in the same star system as Altair VI travelling even a hundred million miles between them would not take a starship long enough for a scheduling conflict. So Altair is out.
That leaves two multiple star systems to choose from. Since Spock mentioned "the Vulcan sun", Vulcan must orbit one star in that star system and the other one or two stars much be far enough away from Vulcan and its star that they appear as stars in the sky of Vulcan instead of as suns.
In 70 Ophiuchi the to stars orbit at distances from 11.4 to 34.8 AU.
The primary star is a yellow-orange main sequence dwarf BY Draconis variable of spectral type K0, and the secondary star is an orange main sequence dwarf of spectral type K4. The two stars orbit each other at an average distance of 23.2 AUs. But since the orbit is highly elliptical (at e=0.499), the separation between the two varies from 11.4 to 34.8 AUs, with one orbit taking 83.38 years to complete.
40 Eridani A is a main-sequence dwarf of spectral type K1. 40 Eridani B and C, are a 9th magnitude white dwarf (spectral type DA4) and an 11th magnitude red dwarf flare star (spectral type M4.5e), respectively. Presumably, while B was a main-sequence star, it was the most massive member of the system, but ejected most of its mass before it became a white dwarf. B and C orbit each other approximately 400 AU from the primary star, A. Their orbit has a semimajor axis of 35 AU (which is the approximate average distance between B and C) and is rather elliptical (eccentricity 0.410).
Since B is a white dwarf that probably once lost a lot of mass in a spectacular way, neither it or its relatively close companion C is likely to have habitable planets.
But A, about 400 AUS from B and C is much more likely to have habitable planets.
An observer on a planet in the 40 Eridani A system would see the B/C pair as unusually bright (magnitudes -8 and -6) white and reddish-orange stars in the night sky. This is not bright enough to diminish the darkness at night, though they would be visible in daylight (assuming an Earth-normal atmosphere). (By comparison, Earth's full moon is magnitude −12.6, and Venus at its brightest is −4.7.)
Thus only 40 Eridani A would look like a sun in Vulcan's sky if Vulcan orbits 40 Eridani A in the habitable zone.
At the nearest distance between 70 Ophiuchi A & B, 11.4 AUs, or about 1.059984 billion miles, a degree of arc would be 9,250,000 miles.
The diameter of the sun is 864,575.9 miles. 70 Ophiuchi A should have a diameter of at least 0.88 of Earth's Sun and thus of at least 760,826.79 miles. Thus it would appear to be 0.0822515 of a degree from 70 Ophiuchi B at their closest.
Normal - Human - visual acuity is about 0.6 of an arc minute, or about 0.00999 degrees of arc. Thus at their closest 70 Ophiuchi A would appear about 8.22 times the minimum width necessary to b e seen as a sun and not as a star.
70 Ophiuchi B has about 0.220 times the luminosity of 70 Ophiuchi A and thus might have only 0.469 times the diameter of 70 Ophiuchi A, but should still be visible as an object and not a star at the closest distance between the two stars, being 3.828 times the minimum necessary width. But 70 Ophiuchi B is cooler than 70 Ophiuchi A and so has to have more than 0.220 times the surface area of 70 Ophiuchi A to emit 0.220 times the radiation as 70 Ophiuchi A.
So 40 eridandi A seems to be the most likely star for Vulcan to orbit around.