TOPIC: Vamping?

When vamping the D chord at frets 2,3,&4 or the G @ frets 3,4,&5 I'll play the 4th string followed by the first, second and third together. My question is when playing a C chord at frets 1 & 2, do I not play string 3 (open)?

When you are playing an open chord (such as a C at the 1st and 2nd frets) you want to avoid playing the open string. Afterall, you can not vamp and open string. You can play the C chord by vamping strings 1 and 2 only. You also should learn to play higher up the neck.

If I play the 1 and 2 string only, I lose the bass note (4 string/2 fret). Is that a correct style vamp for C chord on that part of the neck?

For the most part, that low C is not a good chord to vamp. You can play the 4th string bass (as long as you fret it at the 2nd fret as well) and vamp the 1st and 2nd strings. Do not play or vamp the open string. Furthermore, start learning how to vamp full positions up the neck of the fingerboard.