I've probably thrown 100 different ASP/Access sites at these guys, including full-blown e-commerce sites (yes, they can be done and done well in Access.) If you can write ASP code with a DSNless or OLEDB (ideal) connection string, you will have no problem with these guys.
And no, that's not an affiliate link drop and no I'm not the host. I'm actually a customer who thinks the world of this host. Been with 'em for 5 years now.
My bandwidth and space? Well, I'm on a dedicated box so I've got plenty (225 GB bandwidth and 60 GB space, of which I use around 30-35 GB space and about 50 of the bandwidth.)
The two sites in my signature both have Access backends (although the SEFL one is better one of the two and tends to use it more).
Yeah Tao we do support ASP and MS access. What are your requirements in particular? Have a check at our customizable packages at www.eukhost.com and PM me if you have any queries or doubts regarding them.