At least not with OMP directly. OMP server computes nothing in terms of flight dynamics. The client only "runs" Orbiter and sends its vessels status to the server, only for the sake of signaling to neighbour vessels in other clients. If the client is stopped, all its vessels are deleted.
In OMX, the fork I'm working on, it's quiet similar but if a 1st client is stopped, its vessels are automagically taken over by another client.
If you would run OMP (or OMX) without grahics on one client (orbiter_ng with "D3D9" disabled in the launchpad), and assuming you installed the server and you connected this client to it, then indeed you could "see" the vessels of the 1st client on another client/other PC, but you would still need to pilot the vessels from the 1st client (if it is even feasible without graphics...).
Of course, one can develop a fully new program inspired by OMP-architecture, but it will be a full new complex project, IMO.