This should give you a start:
for f in `cat input.json | jq -r 'keys[]'` ; do
cat input.json | jq ".$f" > $f.json
done
or when you insist on more bashy syntax like some seem to prefer:
for f in $(jq -r 'keys[]') ; do
jq ".[\"$f\"]" < input.json > "$f.json"
done < input.json