I suspect a Y cable is not the answer, because Mic level is very different from typical pickup level. Having said that, I have no idea what a clarinet pickup really is .. is it like a piezo mic or a small condenser mic? Is it unbalanced - similar to a typical electric guitar output?
Have you chosen your FX pedals yet?
You might find that a Line6 Pod X3 or HD might be ideal - simply because (some models) have dual channels (e.g. guitar and vocal), so they are already designed for a Mic and an Instrument to be used simultaneously. The Line6 FX are about as good as anything out there in the pedal range.
Otherwise - I expect you will need to match up levels with a preamp or two. I'm thinking a twin channel preamp will be the way to go. If you use typical guitar stomp pedals you will need to use -10 unbalanced outputs and attenuate to avoid distorting the pedals.
You could use a transformer combiner to mix the two signals together, and also provide some ground loop isolation.
I think there is a lot to be said for using a dual channel Pod that already has all the inputs you need and can digitally mix the two signals, and basically give you full control of FX, signal flow, levels - a controller pedal, tap tempo delays, looping etc, etc ...
Plus - you could run two channels and have stereo FX. Or just keep it mono if the sound guy can't cope with stereo.