Central air unit's fan isn't running and unit is making a fairly load buzzing noise. Blower inside the house is running. Unit is 9 years old.

Sure sounds like the cap to me.

Pull it tomorrow and take it with you to buy a new one.

Most likely the run capacitor. Its a $20 fix. Most caps run both the compressor and the fan. The capacitor terminals will be marked C for common, H for the hermetically sealed compressor, and F for fan.

Get a capacitor with the same voltage and capacitance values or larger.

Yep sounds like the capacitor. Just take a picture with the current wiring setup, get an identical one and swap it out. No problem. $15 part and two minutes of your time.

Loose connection
 

had the same happen to me. started out as teh capacitor, just unhook it and take it with you, also get your model number of unit. they are around $15-$20. might want to take a picture of teh connections before unhooking it. forgot to say that later the fan motor burned out which wasnt but about 80 bucks to replace but it can be a chore to install the new fan depending on how old your unit is. the newer fans have a few more connections we had to route to the panel.

Increased voltage rating is ok, but keep the capacitance values the same.

Looks like it was the capacitor. Old one's top bulged out pretty badly. Part was $20 at Grainger and was switched and tested in less than 10 minutes. I can only imagine what the HVAC guy would have charged me.

Thats the only part they keep in the back of their van. Just tons of capacitors. And they go around changing them out for $120 bucks a pop

They would have charged you a service call ($59 at the place I use but I've seen as high as $99) plus the marked-up capacitor (varies, probably $30-$40).  Some will charge you an additional fee if they do additional work, like checking the freon.  

When I was young, dumb, and hot as hell because the AC wouldn't turn on and therefore not thinking straight, I paid an HVAC company to come find out what was wrong with the AC.  It was just the cap and a little low on freon.  $220.  

You saved yourself about $100 doing it yourself, maybe more if your local HVAC companies are crooks.

I had to deal with this today, I knew something was wrong when I got home from work at 630am and the house read 78 degrees, by noon inside the house it was up to 94 degrees (one of the perks of living in florida ) , and yep a fuckin $11 45uf capacitor was more swollen than someone who ate at a Chinese buffet that used sodium and msg. Funny part. It's a rental house, it eats capacitors and external compressor fans like candy, probably 15 years old, and the owner lives in NY and demands a call authorization for that fix. What a fuckin tightwad. Over a capacitor. Luckily enough we have the cell number of the AC tech guy and it was swapped out in 5 minutes for my elderly dad on dialysis stuck in the house. Now if that cheap bastard in ny would spend 3500 on a new unit he wouldn't be paying all the tech fees every 6 months for the past 10 years.