Can an E-2 visa holder submit a concurrent E-2 visa or transfer to another E-2 visa without leaving the U.S.?

You generally can't get concurrent E-2 visas in the way that you can get concurrent H-1B visas. With regard to transferring an E-2 from one company from one company to another, this is theoretically possible as long as you have been maintaining status on the first E-2. However, I think it might be somewhat difficult in practice unless your first business has been extraordinarily successful and you are now investing these profits in a second business. Speak with an attorney about the specifics of your situation to see how this might work out as a practical matter. Questions such as the level of ongoing involvement in the first business would need to be addressed as well.

Timing is crucial! Often, if you first sell your E-2 to get the funds to start a new E-2 enterprise, the Government can argue that the moment you sold your first business, you were no longer maintaining your E-2 status. Again, timing is critical. You may want to amend your initial petition if you are merely 'adding a new business type' to your initial investement. Speak with a lawyer on this sophisticated issue.