281. Yes, for in-store purchases Costco will only accept American Express credit cards. The key word is credit card though because they do accept debit cards as well.If you purchase from the Costco website however they accept all major credit cards.osts. 281. Yes, for in-store purchases Costco will only accept American Express credit cards. The key word is credit card though because they do accept debit cards as well. If you purchase from the Costco website however they accept all major credit cards.