Q. Am I still able to file for education credits for college if my employer did not reimburse the full $5250 (tax free)?
A. Yes. But, you may only count the tuition paid that was not reimbursed by tax free money, even if the reimbursement was paid in the following year ("I was not refunded for the $6000 spring semester").
By law your employer can only give you $5250 maximum, tax free. . If you got more than $5250, the amount above $5250 is usually already included in box 1 of your W-2 as taxable income. Most students only need $4000 of qualifying expenses to claim the maximum credit.
That brings up the question of why you are just now filing a 2023 return. There's a new urban myth among college students that says they can get a $1000 from the government just for filing a tax form. For most of them, they simply aren't eligible. A full time unmarried student, under age 24, even if you 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 parental 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#en_US_2024_publink53002gd0e674
https://www.irs.gov/instructions/i8863