The HDDs are running out of space on ther server and the N.A. wants to install a USB 2.0 card and an external HDD. Is this a suitable solution? What problems might the network face?

I would get bigger hard drives and clone your drives over using ghost or something like it. In the long run this solution makes more sense and is likely to be less problematic.

I agree with Curt. While there may not be specific problems that we can point to in advance, you really want things as simple as possible on a server. When things go wrong, the problems start to cascade, and the last thing you want is to be in the middle of troubleshooting some obscure problem only to have to deal with a USB drive that loses its connection. The only exception would be if you have basically archive files on the server now... files that you rarely access and are not necessary for the server to boot or do day to day operations, then I would consider moving them to a USB drive or elsewhere on the network.