Our band director has just presented us with a piece that requires of me rapid sixteenth notes of B flat and B natural followed by C back and forth several times. My fingers don't seem to work so quickly. Can anyone offer advice ? I really need it.

Play the Bb (A key + register) and experiment with the top two side trill keys.

oohhhh fun, if i were flute i could manage that, but the equivlent would be c, d and d flat
but that seems to require alot of pinky finger excercise, try practicing doing it slowly and building up the speed till you can do it fo about 4 or 5 bars (or a bar more then you need) at full speed. and find the most comfitable key combination for you rather then leaving the work to one pinky finger (i find c on the top joint and b on the bottom easiest)

practice and experiment with those side trill keys

or alternate fingerings
http://www.wfg.woodwind.org/clarinet/cl_alt_1.html

check the above website for the trill fingerings too as depending upon how fast the tempo is those may work well too.

a full boehm clarinet would make it simple too =-)