There is a strange odor under my house and I am not sure if it is a
moldy smell or a rat smell.  Of course, the rat exterminator wants me
to spend $900 to vacuum up all the droppings.  Should I spend the
money or am I going down the wrong path?

Rats and Mice leave a MUSKY odor.  I would ask the exterminator how he
planned to vaccuum...  I found a document that warns against this,
except under controlled, sterile circumstances.  If I had children, or
outdoor pets, I'd think twice about vaccuuming.

Rats and Mice (read entire)
http://www.in.gov/isdh/regsvcs/saneng/vector/rats_mice.htm#DISPOSAL

Scroll up to read the entire article after reading about "disposal"

~~Cynthia

I would think that there would have to be a LOT of wet droppings under
your house for the smell to reach your nose.  Have you ever been in a
gas station bathroom that had not been cleaned for a while?  That the
smell.

Have you been trying to get rid of rats?  If you have, depending on
the method, the rats may have died and that is want you are smelling. 
I'm not sure about this, but I would think that the smell, if it is
from dead rats, would decrease over a short period of time.

If you have been trying to get rid of rats by killing them with traps,
find the traps and remove them.  Just use rubber gloves and you will
be fine.

If you have been using poison - then the rat carried it back to the
nest and, more than likely, died there.  Find the nest and remove it.