Does the phrase “Payment Due Upon Completion” specify cash or check payment, or does it simply mean payment is required once the work is finished?

No you are not wrong. The sentence is not complete. It could very well mean that when completion is achieved, you have to pay. It does not specify the mode of payment. So you can pay by check, etc. Had the vendor specifically cited what mode of payment (like Cash), only then would you be obligated to pay up in cash.