Do I check the box for my wife having an HSA when it is an HSA account through my work, but both me and wife can use it and have cards issued in our names?

No, Health Savings Accounts (HSAs) are individual accounts. 

Based on your description, I am assuming that you are both covered by your High Deductible Health Plan (HDHP) provided through your work.  If you are both covered under the same plan, it is considered a “family” plan.  Even with a family plan, the HSA associated with that plan would only be in your name, not a joint account.  That means that you are the only one who has an HSA.  This is true even though your spouse has a card to use and can spend the funds in the account. 

In order for your spouse to have her own HSA, she would need to be covered on a separate HDHP through her work or another source.