Possibly, but most likely not
There isn't much exact information about what the suit is comprised of. This may be due to intentional vagueness from the writers.
What we know for sure is that it contains alloys and carbon nanotube based allows:
... After Daredevil's first meeting with Punisher, Potter increased the durability of the suit by adding extra plating and reinforcing the red regions of the suit. Per his recommendation, Potter also incorporated a new carbon nanotube-based alloy which he said was "near indestructible" into a new helmet that included red lenses in the eye holes. He also added advanced alloy gauntlets to the suit's gloves as a bonus. ...
The reasons why I personally believe that it can't be Vibranium, is that Vibranium in the MCU is extremely rare (at least prior to the events of Age of Ultron), with Captain America's shield being the only bit of Vibranium that the US has access to.
Vibranium is also ridiculously expensive, and although Wilson Fisk could probably afford it, a simple blacksmith from Hell's Kitchen probably can't - and remember, Potter created Daredevil's suit pro bono.
I'd say most likely that the suit contains some combination of kevlar and light-weight alloys.
You see, kevlar can protect against bullet damage, as well as cuts and abrasions:
Kevlar is used to manufacture gloves, sleeves, jackets, chaps and other articles of clothing designed to protect users from cuts, abrasions and heat. Kevlar-based protective gear is often considerably lighter and thinner than equivalent gear made of more traditional materials.
Line-X spray is commonly used in truck bed application to strengthen and protect the surface. The company which developed it has recently created a new application of the material called Paxcon which can be used in body armour.
Here's a cool video demonstration of the capabilities of the spray.
Melvin Potter is a mechanic. As such, he could have knowledge of this awesome stuff and may have indeed applied it to parts of the suit (especially the helmet, shin pads and greaves).
This is the MCU after all, and it wouldn't be complete without a bit of "handwaving". We can chalk up this mysterious substance to some sort of unexplained material. It could also mean that the alloys and alloy composites are in fact Vibranium based, just not confirmed to be.
Out of universe
In a recent interview, Ryan Meinerding (one of the costume concept designers of Daredevil (2015)) commented on on the subject, saying:
“The tone that was really communicated was the sense of realism that they were going for”
"When we do these designs, there’s a concept of grounded and a concept of reality."
"The overall layout of that costume is really that it’s meant to look like a Kevlar vest with stuff underneath it. That’s what we were going for."
This aim for realism makes me believe that perhaps Vibranium is just that much too fantastical for this gritty, down-to-earth story.
Additionally, note the reference to "kevlar" armour which solidifies my thinking that the suit probably had some kevlar elements to it.