You can claim reimbursement for travel expenses.

Literal

Trip [for-に] took expenses [topic-は] reimbursement [object-を] can-request.

払い戻しを請求する (to claim a refund/reimbursement) chains two formal words: 払い戻し (refund) + 請求する (to claim/demand). かかった経費 (expenses incurred) uses かかる for costs that arose. できます marks ability/permission.