Yes, but the credit agency will require you to guarantee the conmpany's credit personally, and won't issue a card if they don't like your credit.
If you have sufficient income to pay the debts over time, consider negotiating a payment plan through Consumer Credit Counseling, or filing Chapter 13 bankruptcy. In BK, you file a payment plan to pay the debts in 3-5 years. If the court accepts the plan, the creditors can't charge interest or institute collection proceedings without the court's permission.
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This is not legal advice and you are not my client. Double check everything with your own attorney and your state's laws.
