I have to build a system that accesses a database and
allows certain control procedure to wrok against it, for
example adding and updating customer account information.

I have a skeleton system already in VB.net but have just
been told the application may be used on the internet. A
system of this type is fairly simple would I still be
able to get the functionality from ASP that I can from
VB.NET and ADO.NET? Will I be able to use ASP in Visual
Studio.NE?

Well VB.NET and ADO.NET are significantly different than ASP of old, so
there is a learning curve to consider.
But assuming you have no major time constraints then it would probably be
worthwhile to learn the new stuff.  You can't avoid the future forever - and
the new stuff is truly much better.
A small project is certainly what you would want for your first ASP.NET
application.
I say go for the gusto.

--
I hope this helps,
Steve C. Orr, MCSD
http://Steve.Orr.net