Yes, if you have fangs.
You've already rightly pointed out that you have "appropriate fangs", but I wanted to expound of this a little bit to make sure future askers of this similar question understand that caveat. You need fangs to make the bonus action unarmed strike.
While shifted, you can use your elongated fangs to make an unarmed strike as a bonus action. If you hit with your fangs, you can deal piercing damage equal to 1d6 + your Strength modifier, instead of the bludgeoning damage normal for an unarmed strike. (Wayfinder's Guide to Eberron, p66)
Many longtooth shifters have canine traits that become more pronounced as they shift, but they might instead draw on tigers, hyenas, or other predators.
You retain the benefit of any features from your class, race, or other source and can use them if the new form is physically capable of doing so (PHB p67)
So, yes, you CAN make an unarmed strike in wild-shape, as pointed out in this answer. But here's the catch: you don't have elongated fangs unless you assume a form that has elongated fangs. Wildshape limits your racial features if your new physical form is not capable of doing so. Furthermore, you adopt a new physical form, so you wouldn't have your previously elongated fangs, you would have new teeth. If you transform into a horse, you don't have elongated fangs.
So what this means is that you can't use your bonus action to make an unarmed strike because you don't have the prerequisite (using your elongated fangs). And even if you could, you wouldn't get the elongated fang damage because you also didn't hit "With your fangs".
But lets say you DO have fangs, like the aforementioned tiger. Then you absolutely could make that unarmed strike with fang damage, provided you were Shifted before you made the attack.
What are "Elongated Fangs"?
The "Elongated" fangs are relative to your humanoid form.
Many longtooth shifters have canine traits that become more pronounced as they shift.
So you have sharp canines as a humanoid, but they can't really bite things that well. When you shift, they are elongated. So as long as your new form has fangs, like most predators do, then you can use them. Don't think of elongated fangs as these huge dagger-like, snake-like fangs, but rather just elogated in relation to your humanoid canines (which are already pronounced). A tiger, for example, would have fangs that are elongated in relation to your humanoid fangs. Your DM will decide if your fangs are good enough to use, since the rules don't tell us which animals have fangs nor how long they need to be. We have to use our best judgement.