Do I get the child tax credit from the state of West Virginia this year?

No.  West VA does not have a state level Child Tax Credit. West VA does however over a "Family Tax Credit" which is based on the household size and income levels.  In TurboTax your WVA state forms will automatically include the Family Tax Credit if you qualify based on your entries in the federal interview.  

Here is a link to the WVA Form IT-140 instructions.  You can look  at page 14 to see the tables for the family tax credit.

Basically, each table represents a household size.  Find your income level in the table that applies and use the corresponding percentage to multiply your WVA tax liability.  That will give you the amount of tax credit that will reduce your WVA tax.  Again, TurboTax does this automatically for you.