Can I get credit for tuition fees paid for grandchild that is not my dependent for tax purposes?

No, you cannot claim any education expenses for someone else on your tax return if they are not being claimed as a dependent on your return.

Q.  Can I get credit for tuition fees paid for grandchild that is not my dependent for tax purposes?

A. No.  But, the parents may count the money you paid when calculating the credit, for themselves, if the student is their dependent.  If the student is not  a dependent, he/she may count the money you paid on his tax return, if he otherwise qualifies.  Note: most college age students are not eligible for the refundable tuition credit.*

* A full time unmarried student, under age 24, even if they don't qualify as a dependent, is only eligible for the refundable portion of the American Opportunity Credit if he supports himself by working. You cannot be supporting yourself on family support, 529 plans or student loans & grants. It is usually best if the parent claims that credit. 

Reference: Line 7 instructions for form 8863. https://www.irs.gov/instructions/i8863