I don't know about supply ships, but I believe most of their supplies and mail, etc, came in by train and then had about a day of travel (wagon or walking) to the Camp at Falmouth.
Here's a quote from Walt Whitman:
"WASHINGTON,
January, '63.—Left camp at Falmouth, with some wounded, a few days since, and came here [to Washington] by
Aquia Creek railroad,
and so on Government steamer up the Potomac. Many wounded were with us on the cars and boat. The cars were just common platform ones. The railroad journey of ten or twelve miles was made mostly before sunrise. The soldiers guarding the road came out from their tents or shebangs of bushes with rumpled hair and half-awake look. Those on duty were walking their posts, some on banks over us, others down far below the level of the track. I saw large cavalry camps off the road. At Aquia Creek landing were numbers of wounded going North. While I waited some three hours, I went around among them. Several wanted word sent home to parents, brothers, wives, &c., which I did for them, (by mail the next day from Washington.) On the boat I had my hands full. One poor fellow died going up."
https://iwp.uiowa.edu/whitmanweb/en/writings/civil-war/week-8/down-at-the-front
The Rappahannock next to Falmouth was described by many soldiers as something you could easily wade across, and occasionally did, to trade with the pickets on the other side.
The only boats that seem to have been really useful were flat bottomed Pontoon boats, which were carried overland, and then put in the river to make a bridge. But possibly they could have made it down the river if there hadn't been a war going on.
Even if a boat or a skiff could have made it down the Rappahannock, at least in the Winter and Fall of 1863, it would have been impractical, as the two armies were camped right next to each other for an extended period of time, on either side of the Rappahannock.