Hi Sarah,
Having done a lot of consulting work with bulk email marketers and web hosting firms, I'd say that a VPS account is most likely too weak for the volume you are dealing with. Depends partly on the overall server specs. and VPS software that's running, but we'd still never recommend a VPS for a client with your requirements.
That said, it's hard to suggest a firm solution without knowing more about your company size, budget, etc., however here's some general advice:
Compare the costs of going with a dedicated server (including staff support) and your software vs. using a reputable third-party solution like
Constant Contact.
An advantage of using a reputable third-party firm, besides the ease-of-use and not having to worry about the server(s), is that they proactively manage their email reputation. With the volume of mail that you are sending out, the risks of getting listed on an email blacklist are pretty high, even if you are using permission marketing (double-opt-in).
If the third-party option listed above doesn't sound right for you, or you have the resources to self-manage, I'd tend to agree with your hosting provider that a dedicated server is the way to go.
Good luck, and let us know what you decide!
Rob