Agni is one of the strongest mid laners in the game, and with the incoming items changes and balances, may end up being even stronger.
The reason you, as Vulcan, are struggling so much against him is that Agni has a superior wave clear, a better dash, and a much more consistent damage output, as well as a decent stun.
In short, Agni is just a better mage than Vulcan.
However, if playing carefully, you should be able to hold your own against a compotent Agni.
Use your turret offensively. If Agni dashes in (which isn't uncommon), he has no easy way of getting out of the range of your turret. Place the turret, mark him with your first ability (which means the turret targets him for extra damage) and then try to use your third ability to keep him in the turrets range for as long as possible.
Use your Ult more frequently. As well as being an immensely powerful ability, Vulcans Ult can be used as an easy wave clear in an emergency, a great way to "zone out" the enemy, and an excellent offensive and defensive option. Fire it on yourself if he is too close, or between the two of you if you want him to back off. When combined with the above turret strategy, you could send him back to base quite easily - giving you an easy lane push.
Buy Aegis. A 1-2 second invulnerability will protect you from a lot of Agni's damage potential.
Get a better Jungler. Seriously, the Vulcan/Agni match up is not in your favour, and your Jungler should know this, and be ganking and harassing as much as possible - don't be afraid to ask for help.
If you are still struggling, go to the Solo lane. Despite the current meta seemingly requiring a warrior on the solo lane, a lot of people will play mage in that position, and as such you may be better off swapping with a mage that has a better chance against Agni, such as Chang'e or Aphrodite. Obviously, this is a last resort and will often not be possible.
Despite all this, Vulcan will still have to play well to defeat even a barely competent Agni thanks to the match up being in Agni's favour.