Can I reset Vanity Top onto particle board with residue caulk?
If I understand correctly, you've set a countertop onto a vanity cabinet with silicone, then cut it loose. Some silicone remains on the top edges of the cabinet. You're asking whether you can reset the top with some silicone residue on the cabinet. 

Sure. 

I'd slice as much caulk off the top of the vanity as possible with a fresh utility knife blade, then set the top just as you originally did. The silicone will bond just fine. There's not much stress on it anyway.

If for some reason that doesn't work, screw some 1x2 stock to the inside of the vanity, oriented horizontally and flush on top. (There may be something in the corners already that you can use.) Run 1-1/4" drywall screws through that into the top. That's standard practice for kitchen tops, etc., where a caulk bond isn't needed.