In short, yes you are able to publish a vector tile map using ArcGIS Server as mentioned "Build and publish a vector tile package", "Is there a way to publish vector tiles to ArcGIS Server without Portal?" and "Prerequisites for creating a vector tile service"
Subsequent releases will add the ability for the ArcGIS Runtime to consume vector tiles across all devices (starting with beta releases this fall) as well as the ability for ArcGIS Pro to create and publish vector tiles to ArcGIS Server and ArcGIS Online (planned for early 2016). Source
To go about doing this you will need to use ArcGIS Pro and Portal for ArcGIS as mentioned:
A vector tile package is created in ArcGIS Pro (not supported in ArcMap). You can not publish services from ArcGIS Pro to ArcGIS Server without the use of Portal for ArcGIS.
Starting with release 1.2, ArcGIS Pro allows you to build vector tiles for a map and store them in an easily transferrable vector tile package (.vtpk) file. You can share a vector tile package from ArcGIS Pro to your portal and publish the uploaded vector tile package as a hosted tile layer. Use the resultant hosted tile layer for basemaps in the map viewer or custom applications.
Once you have designated ArcGIS for Server as your portal's hosting server as dicussed here
Once your federated GIS Server has a data store, you can register it as the portal's hosting server.
Sign in to the portal website as an administrator and browse to My Organization Edit Settings > Servers.
From the Hosting Server section of the page, click the drop-down list and choose one of the GIS Server sites you have federated with the portal. If you have added multiple GIS Server sites to your portal, only one can be designated as the hosting server.
Then there are a number of steps to go about publishing as outlined here
Follow these steps to publish a hosted tile layer from a hosted feature layer:
Sign in with an account that has privileges to publish hosted tile layers, and click My Content.
To publish, do one of the following:
Click the hosted feature layer you want to publish and click Publish.
Click Create > Tile Layer and choose a Feature Layer to publish as tiles.
Type a title, tags, and summary, and choose a folder for the hosted tile layer.
Optionally, change the range at which the layer will display by choosing scales from the Visible Range drop-down lists.
Check the boxes next to the scales at which you would like to generate tiles.
Then click Create Tiles.
Review the message about how many tiles will be created. If you still want to proceed, click Yes, create tiles.