Not directly. We are a shared hosting provider. Credit card issuers impose security requirements on accepting credit cards that prevent you from requesting or storing credit card information on any shared server unless very specific criteria ("PCI DSS requirement A.1") exist and have been audited. It is our opinion that the criteria as written require VPS-type services, which we are not.
Therefore to accept credit cards and most other payments, you will need to use a third-party secure checkout service. The most common services all work with our service, and we don't have any particular recommendations.