Can you do a chargeback on a debit card purchase? The card apparently has a Visa logo on it. It looks like we may be getting screwed by a vendor.

You should contact your card issuing bank to be sure about this.

Can you do a chargeback on a debit card purchase? The card apparently has a Visa logo on it. It looks like we may be getting screwed by a vendor.
If you are in the U.S., there are laws governing this. For example, the issuing bank MUST investigate your complaint. I'm in California, and I had HostPro charge my card, and they wouldn't issue a refund. Per California law, the bank conducted an investigation, and since they did not have a resolution within 5 business days, they had to allow me PROVISIONARY use of the funds. I think they have up to 30 to make a final decision.
You should give your bank a call, as Chicken suggested. It should be pretty painless.

It's really up to the bank for judgement with debit cards. My bank did a chargeback for me because I sent back an item for refund and all correspondance to the place halted once I proved they received it. After 30 days of no response, a dead phone number, and them having both the product & my money I called up the bank. A few days later the money was back in my account. So yes, you can do a chargeback as long as there's good enough reasoning for your bank.