Of course. We've seen it done several times in THG, DS9 and Voyager. The big problem with transporting at warp is precisely matching warp velocities, as explicitly noted in TNG. A ship is automatically matching its own velocity...
Sorry when you say site to site - do you mean aka - like from Engineering -> Main Bridge - if so, of course you can, because, it's all within the warp field, which is 'space-normal'. The only exception could be if you were going at Max warp, in which case power could be a problem!