If I am the custodial parent, and no exemption was given, (due to an unofficial agreement between unmarried parents), do I qualify for the Oregon Kids Credit?

It depends.  If you meet the income levels (less than $30,000) and the age of the child (0 to 5) meets the criteria and the child lived with you for more than 1/2 the year then,  yes.  TurboTax will figure the credit based on your dependent entries in the "My Info" section and your income entered in "Wages & Income".  

Here is a link to an OR tax page discussing several OR credits available to families.