Sql Server - How To Create New Database Programmatically In MVC Application

I have worked on a timesheet application application in MVC 2 for internal use in our company. Now other small companies have showed interest in the application. I hadn't considered this use of the application, but it got me interested in what it might imply.

I believe I could make it work for several clients by modifying the database (Sql Server accessed by Entity Framework model). But I have read some people advocating multiple databases (one for each client).

Intuitively, this feels like a good idea, since I wouldn't risk having the data of various clients mixed up in the same database (which shouldn't happen of course, but what if it did...). But how would a multiple database solution be implemented specifically?

I.e. with a single database I could just have a client register and all the data needed would be added by the application the same way it is now when there's just one client (my own company).

But with a multiple database solution, how would I create a new database programmatically when a user registers? I have done all database stuff using Linq to Sql, and I am not very familiar with regular SQL programming...

how this could be done (as well as input on whether it is a good idea or if a single database would be better for some reason).

EDIT:

I have also seen discussions about the single database alternative, suggesting that you would then add ClientId to each table... But wouldn't that be hard to maintain in the code? I would have to add "where" conditions to a lot of linq queries I assume... And I assume having a ClientId on each table would mean that each table would have need to have a many to one relationship to the Client table? Wouldn't that be a very complex database structure? As it is right now (without the Client table) I have the following tables (1 -> * designates one to many relationship):

Customer 1 -> * Project 1 -> * Task 1 -> * TimeSegment 1 -> * Employee

Also, Customer has a one to many relationship directly with TimeSegment, for convenience to simplify some queries.

This has worked very well so far. Wouldn't it be possible to simply have a Client table (or UserCompany or whatever one might call it) with a one to many relationship with Customer table? Wouldn't the data integrity be sufficient for the other tables since the rest is handled by the relationships?