Do I have to claim my 18 yr old daughter (student and supported by me) on my tax return or can she file on her own to receive the education credit?

Q. Do I have to claim my 18 yr old daughter (student and supported by me) on my tax return?

A. No, but there is seldom a reason for you not to.  Even if you don't, she cannot "claim herself".

Q. Can she file on her own to receive the education credit?

A. Simple answer: No. 

While technically there is a provision that allows your student-dependent to claim a federal tuition credit, from a practical matter it seldom works out.  A student, under age 24, is only eligible for the refundable portion of the American Opportunity Credit (AOTC) if he/she supports himself by working. She cannot be supporting herself on student loans & grants and 529 plans and parental support (as you indicated she is).  It is usually best if the parent claims that credit.  

If the student actually has a tax liability, there is a provision to allow him to claim a non-refundable tuition credit. But then the parent must forgo claiming the student as a dependent, and the $500 other dependent credit.  The student must still indicate that he can be claimed as a dependent, on his return. This is worth up to $2500 (AOTC shifts to all non refundable)