VB.NET & databases

I am about to start working on a front-end application to write to, read from, and modify records in a centrally located database. In the past, I've used VB 6 & the ADO control with a MS Jet database. I need to have very large # of records, and a unique number assigned to each record. (ie. a user from Texas may a unique # of "TX-000-101" The next user from texas would need to automatically have a user # of "TX-000-102". What type of database would be best suited from my project? Also, would I be able to implement this as one database, or would there have to be a database for each state. Anything to point me in the right direction is much appreciated.

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What type of database would be best suited from my project?

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A client-server relational one. My vote would be for SQL Server (or MSDE if you don't have funds for a full db license)

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Also, would I be able to implement this as one database

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Absolutely - make 'State' one of your table key fields.
Do a google for Dr E. F. Codd for a background if you've not done any relational database stuff before.

I've dealt mostly with Access Databases in the past (as well as jet). I'd like to possibly use MySQL as it is open source. Will I have difficulties interacting with the database via VB if I use MySQL?

You can use Access, if you want. It's a honest and simple DB! I used MSDE and Access. Access act as a file server, so your LAN will probably have to work harder than using MSDE. MSDE acts as Client\Server, so your server will probably have to work harder than using Access. Access is weaker with transaction, but very simple to administrate. I started using Access and ADO, not ADO.Net. Now I'm converting some my programs to use ADO.Net. It's not very far if you use a connected way. Ah.... remember that ADO.Net is mainly designed to work disconnected and, anyway, it doesn't implement a pessimistic locking. In most application, it is not a big problem, but you have to face the fact that the record you are editing on your PC, could be cahnged by other users on your server located DB!
I've not an experience with MySQL, but I think you will not have problem... there are many developers that use it and many tools, I guess.
Anyway, if your is a professional application, probably the problem should be which is the best DB, not few hundreds of dollars to buy it. For example, MSDE is free and it works with SQL Server core. You can easily upgrade it to SQL Server when you exceed 2 Giga of DB or when there are often more than 5 users connected to DB at the same time. In this scenario, probably you can spend a little of money to buy a license!
Good job! :)