It sounds as though this organization needs to do some serious work on its financial controls and governance practices, not only to protect the organization but to protect the board Chair as well.
There are no absolute rules about who should have a nonprofit’s credit card, but whoever carries one should use it only for proper organizational purposes.  If your board chair is using the card for personal expenses, those funds will be considered an “automatic excess benefit” unless the organization treats the amounts as compensation in the year spent.  That means reporting the amounts on your Form 990 tax information return and having the Chair report them on his/her personal income tax and paying the taxes.  If the expenses are not fully justified as organizational expenses and they are not treated as compensation, the IRS can charge a 25% excess benefit tax on the Chair, and a 200% tax on the Chair if the funds are not restored.  There are additional fees for late reporting the automatic excess benefit.  Any manager who knowingly approves such excess benefit may also face a penalty tax.  (See Ready Reference Page: “IRS Issues Tips to Agents on Collecting ‘Automatic’ Excess Benefits Taxes.”)
In addition, if the amounts are not treated as compensation, an aggressive district attorney or U.S. Attorney could consider it theft and prosecute criminally.Even if the Chair treats the expenses as personal income, the organization should determine that the compensation is reasonable for the services rendered so that it is not considered an excess benefit because it is too much.It doesn’t sound as though the organization is carefully looking at this issue.  You ought to develop strict policies on the use of credit cards, establish checks and balances so that the Chair or any other user can’t bully someone into approving the charges, and, along with your auditor, review compliance on a regular basis.  If you don’t, the failure could easily come back to haunt you all.