The pump should run but will not pump water. I would also imagine that water could get into the pumps electrics which is not a good thing.
I am a great believer in fixing things, but over the years have decided that its a waste of time when it comes to something like a pump. If the pump has worked well and given good service, then replacing the broken part will extend the units life until the next thing wears out or breaks, ie a bearing or carbon brushes. You just end up throwing good money after bad and end up with unreliable tools to do your job.
Just replace the pump with a new one if it is the diaphragm thats gone.
On the other hand, a few people have got caught out with frozen pumps during winter and the water expansion inside the pump has dislodged the diaphragm. In that case I would consider my options were I to replace the diaphragm. ie cost and downtime as well as any other damage, ie cracked housing.
