This may be a dumb question but...This concept I do not have much experience in.

We have four offices (B, C, D, & E) that connect to a private network (network A).
Each site has a Cisco router that ties back to the private network. They filter our web content and allow one of the applications we use connect to the network at network A.
We have a T1 line at office B, C, & D and at office E we have DSL.
My question is. How can we put a locally provided internet service in place and still route back to network A so when that application is used it accesses the necessary network.

I wanted to get the internet setup at each location and VPN to office B then and have every funnel in to Network A anytime that application is accessed. But I have never configured that and I am afraid that the bandwidth would come to a standstill.
The DSL connection would be 3mpbs down and 512 up at each site. And the connection to Network A at the main site is a full T1. There are about 60 nodes connecting on that T1 now, but since it will be just a few accessing it when they need to use the application I figure it wont be too bad of a connection but since the VPN connection is running already with a low upstream I don't know what the overall connection would be like.
 

I don't think anyone here would be able to answer your question as to whether your application would saturate your network, that's an answer only you can find with some monitoring and graphing of connections in your sites.

To the idea of putting internet at all the sites instead of the private links and VPN'ing back in, this is common and a lot of people do it to save money.

Do you not want your main office to proxy/filter the internet from the branches anymore?  If you want them to use their own internet connection but VPN in to the main office for traffic destined to that office you would use split tunneling.

It's easy to do and very common either way.