Encryption and signatures are two distinct activities, and the difference matters a lot in that respect.
Encryption is about maintaining data confidentiality, except for "authorized users". When these users have private keys, encryption is said to be "end-to-end" in that the decryption will occur on the recipient's device, and can be done only with the help of a private key made accessible to that device by its owner. Corporations, generally speaking, don't like that at all. Encryption prevents them from performing some automatic processing, in particular antivirus scanning (for incoming emails) and detection of information leakage (for outgoing emails). Moreover, encryption raises the issue of data loss: if the private key becomes unavailable, the encrypted emails are no longer readable, since they are typically stored encrypted. Notably, when the user becomes unavailable (he was fired; he retired; he was killed in an accident;...) then the new tenant of his function in the organization must be able to read the corporate emails which were received by that previous user.
Therefore, in a business context, it is often preferred when encryption is centralized -- if there is encryption at all. Emails would often be stored "as is" on servers for which security is done on an "intrusion prevention" basis, rather than mathematically (with cryptography). Encryption is used for data in transit (typically SSL). Users authenticate with the central server when reading or sending emails, but this does not necessarily entails a RSA private key in a smart card. When an email is sent to an external recipient (outside of the organization) then, even if encryption is to be applied, it will be done on a dedicated gateway system, after all due analysis systems have inspected the outgoing email contents and have given the go for further email propagation.
Signatures are avoided. Strictly speaking, a signature is a legal weapon that you build and then point toward yourself. The point of the signature is it is a proof, convincing for third parties, that the email contents are indeed what the signer sent. This can only serve as a way to coerce the signer into accepting extra responsibilities.
If you think properly, then you do not want to sign. You want other people to sign what they send to you. So business try to avoid signatures, and will deploy a system for signing emails only if it is to replace another pre-existing system where there already are signatures done in some other way which is even more inconvenient.
This sums up as the following: if you want to support end-to-end encryption for emails, and/or email signatures, then... think again. Chances are that it is not that good an idea.
Private keys on smart cards, though, are deployed for authentication. Users receive smart cards (possibly with USB token format) containing private keys, and they use that for client SSL connections with a Web server; the client is then their Web browser. This will be for access to sensitive data, so, generally, the corporation will ban such access from "home devices": the employee is not supposed to connect from his own iPad. He shall use only the approved, company-provided laptop computer that is riddled with company-approved antivirus software, and suitably "locked down".
If the daily business calls for some signatures (e.g. medical prescription, in a world of computerized medical file: physicians must be accountable for what they prescribe), then this will not be a basic email, but a strictly formatted "message", signed through a local specific application or some similar trickery (ActiveX control or signed Java applet in a Web site). There again, home devices will not be considered acceptable.
All that being said, some companies have tried, over the years, to sell "virtual smart cards" which usually boil down to some secret value (e.g. a RSA private key) stored with password-based encryption on a server. To a large extent, a GnuPG keyring is an instance of such a system: contains a private key, protected through encryption with a password; put a copy of that keyring on a backup server, and you are all set. "Virtual smart card" commercial solutions mostly aim at making this process easier and more integrated with existing applications (in particular Outlook and Internet Explorer).
To my knowledge, none of such systems was a true commercial success, for the reasons explained above: the added value of these "virtual smart cards" is for situations which you would like, company-wise, to avoid altogether.