How close is "close?"

Beyond that, I seriously doubt it will interfere with grape roots.

Lungwort, Pulmonaria, is a low growing hardy perennial with fibrous roots. It will have no effect on your grape vine but the 2 like different conditions. Pulmonaria likes a position in dappled shade and grapes need sun. Also being perennial Pulmonaria might make it awkward to weed or feed around your grapes. If you want maximum fruit yield I'd keep the base of the grapes clear. If you don't expect optimum grape crops and just want an attractive bed then stick in some Pulmonaria. The grapes themselves could provide the dappling.

Thank you all for your answers. I'm planning the lungwart in a separate, but close, bed to the grapes. The grapes will be providing some shade to the lungwart, but the lungwort will not be planted directly under the grapes.