This is obviously not legal.  Your sister has engaged in a fraudulent act in pretending to be you.  You can report her to the police and have her prosecuted.  You could also sue her in civil court for any damages that you incur as a result of her acts.  You should also contact your credit card company to prove to them that it was your sister that closed the account and that you want it reopened.   If you file a police report, you should send that to the credit card company as proof that you contend that your sister closed the account.  You should also change any password that someone close to you like your sister could figure out.   This is often how family members cancel or use their relative's accounts.   Use passwords that have no relation to your birthdays, schools, names   . . .  etc.      Good luck.

You can take the extreme action of calling the police and telling them that your sister is misrepresenting herself to be you and closing accounts that you rely on for your business, which cost you money.  The police may arrest her and possibly charge with with fraud.  You may also sue your sister for tortious interference with a contract.  In this case, your sister is interfering with your contract with the credit card company and is costing you money as you are not able to run your business effectively and making charges/purchases for the business.  I personally suggest that you hire a lawyer to write a letter to the credit card company regarding your sister's misrepresentation and then a letter to your sister warning her that you will sue her or file charges against her if this happens again.