I'm looking to film some documentary style footage. Does a shotgun mic work alright for close applications, such as an alternative to expensive lav micing? I know there is often a distortion problem on shotgun mics though.
What are some cheap but still decent shotgun mics?
Or should I stick with a handheld?

I'm going into this with no audio equipment, and low budget. I do have substantial sound experience, however.

I am by no means an audio expert (in fact, I'd say I'm at the opposite end of the spectrum--ha-ha!) but I can tell you that for Doc-style interview work, I do not like the sound captured by my shotgun mics. I have five different mics, and each is best suited for a farily specific purpose. But for straight-on interview work I generally use my wireless lav mic, a Sony UWP-C1 (62); of all my mic choices, it picks up the cleanest and most accurate interview vocals--but admittedly, almost all my work is outdoors, so your use and mileage may vary. Now if I'm my home studio, or if I'm interviewing indoors (a rarity for me) in a room without too much noise or poor acoustics, I'll use my MXL 990, but it's really not designed for that sort of application. But for general all-purpose interviews, I prefer the wireless lav.

Now granted, if you can only have ONE mic, and you're on an extremely tight budget, you should consider a Rode VideoMic (1/8" mini) or a Rode NTG-2 (XLR) shotgun mic...and then shoot an interview with it and see if you can live with the sound. Depending on how demanding your audio needs are, it *might* suit your purposes. At the very least, the Rode mic will work for all of your non-interview work.

2¢ (and taken with a grain of salt)

What do those run typically?

I'm looking to film some documentary style footage. Does a shotgun mic work alright for close applications, such as an alternative to expensive lav micing? I know there is often a distortion problem on shotgun mics though.
What are some cheap but still decent shotgun mics?
Or should I stick with a handheld?

I'm going into this with no audio equipment, and low budget. I do have substantial sound experience, however.

If you can only afford ONE mic, I suggst you consider a hypercardioid rather than a shotgun. Shotguns add a great deal of colouration to the sound when used in a reflective environment such as indoors that hypers do not. The upshot is that hypers work well both indoors and out while shotguns only work well out of doors or in similarly non-reflective spaces. Thus the hyper gives you more bang for the buck.

I'm not quite sure why you say "expensive lav micing" though. Good quality lavs are not THAT expensive. Especially if you don't have a boom operator to get a shotgun or hyper close to the subject, a conventional wired lav on the subject will give you much better sound.