What spinning rod, $100 or less, would you buy for fishing top-water frogs like the scum frogs in heavy cover?

I would highly suggest baitcast for frog fishing a high speed reel and you will land more fish.  JMO tho

I've noticed that a lot of attractive rod designs are not offered in "heavier" ratings, maxing out at #14 or 5/8 oz.  

Fenwick has some spinning rods that are rated up to 17 or 20 lb test.  The HMX is well under $100 and occassionally the HMG will be on sale for <$100.

Check out the Red Bone rods made by Hurricane.  They are about $90 and are good for the mney.  I have 7.5 or 8' I use with braid for saltwater snook fishing and C-rig.  I know Sports Authority carries them but they may not have that size.

Personally unless you are investing in Salt Water spinning tackle I would not try using it for fishing frogs in heavy cover.  

A medium heavy rod with a fast action is the bare minimum in my opinion.  Also, a reel with a high/hyper gear ratio comes in handy.  Because of this I would seriously consider baticasting gear for frogs.  As lunkerfever said, you are going to end up with more fish in the boat.