“Can I use my canvas gun sling for Picket Post?"
No.  They did not exist yet.  The Richmond Arsenal ordered them into production in May 1864 and it took a bit to actually get them contracted for, made, and issued.  Use them at Appomattox.
"Did they have gun slings before then?  What kind?"
By the time we're portraying at Picket Post, the Richmond Arsenal had issued out over 40,000 British-made gun slings.  But moreso than that, they issued out even more gun slings made from pre-cut kits run through the blockade, or using the tens of thousands of gun sling buckles that were included in blockade runner cargos.
If you've got a hankering to use a gun sling, either on your Enfield or even your .58 Springfield rifle musket (which is actually the preferred weapon for our 17th Virginia impression), we have some nice waxed flesh CS/"British" gun slings available.  The original might actually be a CS-produced piece, as the location the adjustment holes are punched allow it to attach to an Enfield, but also includes some holes that will not work with an Enfield, but will allow it to just barely fit a Springfield.