Do I need to add supports before replacing a cripple wall stud?
It probably would not sag enough to do any damage, but "probably" isn't good enough.  

Without a picture it's hard to figure exactly what you have from your description - I can't picture a cripple wall without a top plate.  This is a typical cripple wall framing:  



One way to build temporary support is to build a temporary stud wall a few inches in front of the wall you're working on, with the top plate supporting the joists.  You can wedge some shims between the top plate and the joists for a snug fit.  You can arrange the studs in the temporary wall so that it's easy to reach past the temporary wall to get to the sill and studs you need to work on.  

Another way is to lag a 2x10 across the face of the studs, spanning two studs that won't be disturbed on either side.  If you have 40" between the studs that won't be disturbed, and 16" to the studs on either side, an 8' 2x10 would work.  This lets you cut the bottoms off the studs over the rotten sill, repair the sill, install a temporary stud between the studs you cut, remove the 2x10, remove the rest of the studs you cut, and finally replace the studs you removed.   

A reciprocating saw will work, but an oscillating tool with a blade that can cut nails is very handy for cutting out the sill and studs.